A few example designs and data sets for this module are available in the R package apts.doe, which can be installed from GitHub

library(devtools)
install_github("statsdavew/apts.doe", quiet = T)
library(apts.doe)

References will be provided throughout but some good general purpose texts are

• Atkinson, Donev and Tobias (2007). Optimum Experimental Design, with SAS. OUP
• Wu and Hamada (2009). Experiments: Planning, Analysis, and Parameter Design Optimization (2nd ed.). Wiley.
• Morris (2011). Design of Experiments: An Introduction based on Linear Models. Chapman and Hall/CRC Press.
• Santner, Williams and Notz (2019). The Design and Analysis of Computer Experiments (2nd ed.). Springer.

These notes and other resources can be found at https://statsdavew.github.io/apts.doe/

• Observational studies
• Sample surveys
• Designed experiments

Why do we experiment?

• key to the scientific method
  (hypothesis – experiment – observe – infer – conclude)

• potential to establish causality …

• … and to understand/improve complex systems depending on many factors

• comparison of treatments, factor screening, prediction, optimisation, …

Design of experiments: a statistical approach to the arrangement of the operational details of the experiment (eg sample size, specific experimental conditions investigated, …) so that the quality of the answers to be derived from the data is as high as possible.

1. Multi-factor experiment in pharmaceutical development.

Key to developing new medicines is the identification of optimal and robust process conditions (e.g. settings of temperature, pressure etc.) at which the active pharmaceutical ingredient should be synthesized.

[Somewhat confusinging, the FDA refer to this as identification of a “design space”.]

An important step in is this methodology is a robustness experiment to assess the sensitivity of identified conditions to changes in all (or at least very many) controllable factors.

While developing a new melanoma drug, GlaxoSmithKline performed an experiment to investigate sensitivity to 20 factors. Their experimental budget allowed only 10 individual experiments (runs) to be performed.

2. Optimal design to calibrate a physical model.

Physical (mechanistic, mathematical, …) models are used in many scientific fields. Typically, they are derived from fundamental understanding of the physics, chemistry, biology …

Most commonly, these models are solutions to differential equations. The models usually contain unknown parameters that should be estimated from experimental data.

Biologists at Southampton were studying the transfer of amino acids between mother and baby through the placenta. They could control the times at which observations were taken and the initial concentrations of amino acids (see Overstall, Woods, and Parker 2019).

3. Computer experiments to optimise ride performance in luxury cars

Suspension settings can be used to improve the ride performance in cars. Optimising settings across many different car models would take many hundreds of hours of testing, so computer simulations are used.

Jaguar-Land Rover wanted to find suspension settings robust across different car models using a computer experiment (KTN workshop).

Consider an experiment to compare two treatments (eg drugs, diets, fertilisers).

We have \(n\) subjects (eg people, mice, plots of land), each of which can be assigned to one of the two treatments.

A response (eg protein measurement, weight, yield) is then measured from each subject.

Question: How should the two treatments be assigned to the subjects to gain the most precise inference about the difference in expected response from the two treatments.

Assume a linear model for the response \[ y_i = \beta_0 + \beta_1 x_i + \varepsilon_i\,,\qquad i=1,\ldots,n\,, \] with \(\varepsilon_i\sim N(0, \sigma^2)\) independently, \(\beta_0,\beta_1\) unknown parameters and \[ x_i = \left\{ \begin{array}{cc} -1 & \mbox{if treatment 1 is applied to subject $i$}\,, \\ +1 & \mbox{if treatment 2 is applied to subject $i$} \end{array} \right. \] The difference in expected response between treatment 1 and 2 is \[ E(y_i\,|\, x_i = +1) - E(y_i\,|\, x_i = -1) = \beta_0 + \beta_1 - \beta_0 + \beta_1 = 2\beta_1 \] So we need the most precise possible estimator of \(\beta_1\)

Both \(\beta_0\) and \(\beta_1\) can be estimated using least squares (or equivalently maximum likelihood).

Writing \[ \boldsymbol{y}= X\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\,, \] we obtain estimators \[ \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{y} \] with \[ \mbox{Var}(\hat{\boldsymbol{\beta}}) = \left(X^\mathrm{T}X\right)^{-1}\sigma^2 \] In this simple example, we are interesting in estimating \(\beta_1\), and we have \[ \begin{split} \mbox{Var}(\hat{\beta_1}) & = \frac{n\sigma^2}{n\sum x_i^2 - \left(\sum x_i\right)^2}\\ & = \frac{n\sigma^2}{n^2 - \left(\sum x_i\right)^2} \end{split} \]

Hence, we need to pick \(x_1,\ldots,x_n\) to minimise \(\left(\sum x_i\right)^2 = (n_1 - n_2)^2\)

• denote as \(n_1\) the number of subjects assigned to treatment 1, and \(n_2\) the number assigned to treatment 2, with \(n_1+n_2 = n\)
• it is obvious that \(\sum x_i = 0\) if and only if \(n_1 = n_2\)

Assuming \(n\) is even, the “optimal design” has \(n_1 = n_2 = n/2\)

For \(n\) odd, let \(n_1 = \frac{n+1}{2}\) and \(n_2 = \frac{n-1}{2}\)

We can assess a designs, labelled \(\xi\), via its efficiency relative to the optimal design \(\xi^\star\): \[ \mbox{Eff($\xi$)} = \frac{\mbox{Var}(\hat{\beta_1}\,|\,\xi^\star)}{\mbox{Var}(\hat{\beta_1}\,|\,\xi)} \]

n <- 50
eff <- function(n1) 1 - ((2 * n1 - n) / n)^2
curve(eff, from = 0, to = n, ylab = "Eff", xlab = expression(n[1]))

• Treatment – entities of scientific interest to be studied in the experiment
  eg varieties of crop, doses of a drug, combinations of temperature and pressure

• Unit – smallest subdivision of the experimental material such that two units may receive different treatments
  eg plots of land, subjects in a clinical trial, samples of reagent

• Run – application of a treatment to a unit

Fabrication of integrated circuits (see Wu and Hamada 2009)

• an initial step in fabricating integrated circuits is the growth of an epitaxial layer on polished silicon wafers via chemical deposition

Unit

• set of six wafers (mounted in a rotating cylinder)

Treatment

• combination of settings of the factors
  • A : rotation method (\(x_1\))
  • B : nozzle position (\(x_2\))
  • C : deposition temperature (\(x_3\))
  • D : deposition time (\(x_4\))

\[ y_{ij} = \tau_i + \varepsilon_{ij}\,,\qquad i=1,\ldots,t;\,j=1,\ldots,n_i\,, \] where

• \(y_{ij}\) : measured response from the \(j\)th unit to which treatment \(i\) has been applied

• \(\tau_i\) : treatment effect (expected response from application of the \(i\)th treatment)

• \(\varepsilon_{ij}\) : random deviation from the expected response [typically \(\sim N(0,\sigma^2)\)]

The aims of the experiment are achieved by estimating comparisons between the treatment effects, \(\tau_k - \tau_l\).

Experimental precision and accuracy are largely obtained through control and comparison.

Three key model assumptions are:

• additivity (response = treatment effect + unit effect)
• constancy of treatment effects (treatment effect does not depend on the unit to which it is applied)
• no interference between units (the effect of a treatment applied to unit \(j\) does not depend on the treatment applied to any other unit)

See Dasgupta, Pillai, and Rubin (2015) for discussion of these assumptions for factorial experiments

Stratification (blocking)

• account for systematic differences between batches of experimental units by arranging them in homogeneous sets (blocks)
  • if the same treatment was applied to all units, within-block variation in the response would be much less than between-block
  • compare treatments within the same block and hence eliminate block effects

Replication

• the application of each treatment to multiple experimental units
  • provides an estimate of experimental error against which to judge treatment differences
  • reduces the variance of the estimators of treatment differences

Randomisation

• we randomise features such as the allocation of units to treatments, the order in which treatments are applied, …
  • protects against lurking (uncontrolled) variables (model-robust) and subjectively in the allocation of treatments to units

Randomisation is perhaps the key principle in the design of experiments

• it protects against model misspecification (bias), and hence allows causality to be established
  • a clear difference between treatments can only be an accident of the randomisation or a consequence of the treatments
• unbiased estimation of \(\tau\) and \(\sigma^2\), even if the errors are not normally distributed
• exact tests for differences between treatment effects are available (Basu 1980)

Without randomisation, unobserved confounders (\(U\)) can induce a dependency between
the response (\(Y\)) and treatment (\(T\)) cf Cox and Reid (2000), p.35

With randomisation, unobserved confounders (\(U\)) are independent of the treatment (\(T\)). Marginalisation over \(U\) does not induce an edge between \(T\) and \(Y\) cf Cox and Reid (2000), p.35

Fabrication of integrated circuits (Wu and Hamada 2009, p155)

Treatment

• combination of settings of the factors
  • A : rotation method (\(x_1\))
  • B : nozzle position (\(x_2\))
  • C : deposition temperature (\(x_3\))
  • D : deposition time (\(x_4\))

Assume each factor has two-levels, coded -1 and +1

Each factor has two levels \(x_k = \pm 1,\, k=1,\ldots,4\)

A treatment is then defined as a combination of four values of \(-1, +1\)

• eg \(x_1 = -1, x_2 = -1, x_3 = +1, x_4 = -1\)
• specifies a setting of the process

Assume each treatment effect is determined by a regression model in the four factors, eg \[ \tau(\boldsymbol{x}) = \beta_0 + \sum_{i=1}^4\beta_ix_i + \sum_{j=1}^4\sum_{i>j}^4\beta_{ij}x_ix_j \]

with(cirfab, cirfab[order(x1, x2, x3, x4), ])

##    x1 x2 x3 x4     ybar
## 2  -1 -1 -1 -1 13.58983
## 1  -1 -1 -1  1 14.59000
## 4  -1 -1  1 -1 14.04983
## 3  -1 -1  1  1 14.24000
## 6  -1  1 -1 -1 13.94000
## 5  -1  1 -1  1 14.65000
## 8  -1  1  1 -1 14.14017
## 7  -1  1  1  1 14.40000
## 10  1 -1 -1 -1 13.72000
## 9   1 -1 -1  1 14.67000
## 12  1 -1  1 -1 13.90000
## 11  1 -1  1  1 13.84017
## 14  1  1 -1 -1 13.87983
## 13  1  1 -1  1 14.56000
## 16  1  1  1 -1 14.11017
## 15  1  1  1  1 14.30000

• treatments in standard order

• \(\bar{y}\) - average response from the six wafers

\[ \boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\varepsilon}\,,\qquad \boldsymbol{\varepsilon}\sim N(\boldsymbol{0}, \sigma^2I)\,,\qquad \hat{\boldsymbol{\beta}} = \left(X^\mathrm{T}X\right)^{-1}X^\mathrm{T}\boldsymbol{Y} \]

• model matrix \(X\) has columns corresponding to intercept, linear and cross-product terms

• information matrix \(X^\mathrm{T}X = nI\)

• regression coefficients are estimated by independent contrasts in the data

cirfab.lm <- lm(ybar ~ (.) ^ 2, data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

Main effect of \(x_k\): \[ [\text{Avg. response when $x_k = 1$}]\, -\, [\text{Avg. response when $x_k = -1$}] \]

Interaction between \(x_j\) and \(x_k\): \[ [\text{Avg. response when $x_jx_k= 1$}]\, -\, [\text{Avg. response when $x_jx_k = -1$}] \]

Higher-order interactions defined similarly

Assuming -1,+1 coding, there is a straightforward relationship between factorial effects and regression coefficients

• main effect of \(x_k\) is equal to \(2\beta_k\)
• interaction between \(x_j\) and \(x_k\) is equal to \(2\beta_{jk}\)

Using the effects package:

library(effects)
plot(Effect("x1", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x2", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x3", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)
plot(Effect("x4", cirfab.lm), main = "", rug = F, ylim = c(13.5, 14.5), aspect = 1)

plot(Effect(c("x3", "x4"), cirfab.lm), main = "", rug = F, ylim = c(13.5, 15), 
     x.var = "x4")

\(X^\mathrm{T}X = nI \Rightarrow \hat{\boldsymbol{\beta}}\) are independently normally distributed with equal variance

Hence, we can treat the identification of important effects (ie large \(\beta\)) as an outlier identification problem

• plot (absolute) ordered factorial effects against (absolute) quantiles from a standard normal
• outlying effects are identified as important

Cuthbert (1959)

Using the FrF2 package

library(FrF2)
par(pty = "s", mar = c(8, 4, 1, 2))
DanielPlot(cirfab.lm, main = "", datax = F, half = T)

An unreplicated factorial design provides no model-independent estimate of \(\sigma^2\) (Gilmour and Trinca 2012)

• any unsaturated model does provide an estimate, but it may be biased by ignored (significant) model terms
• this is one reason why graphical (or associated) analysis methods are popular

Replication also increases the power of the design

• common to replicate a centre point
• allows a portmanteau test of curvature

Refit a linear model to the fabricated circuits experiment excluding interactions.

cirfab2.lm <- lm(ybar ~ (.), data = cirfab)
coef(cirfab.lm)

##  (Intercept)           x1           x2           x3           x4        x1:x2 
## 14.161250000 -0.038729167  0.086270833 -0.038708333  0.245020833  0.003708333 
##        x1:x3        x1:x4        x2:x3        x2:x4        x3:x4 
## -0.046229167 -0.025000000  0.028770833 -0.015041667 -0.172520833

coef(cirfab2.lm)

## (Intercept)          x1          x2          x3          x4 
## 14.16125000 -0.03872917  0.08627083 -0.03870833  0.24502083

cbind(sigma1 = summary(cirfab.lm)$sigma, df1 = summary(cirfab.lm)$df[2],  
      sigma2 = summary(cirfab2.lm)$sigma, df2 = summary(cirfab2.lm)$df[2])

##        sigma1 df1    sigma2 df2
## [1,] 0.137152   5 0.2396108  11

Effect sparsity

• the number of important effects in a factorial experiment is small relative to the total number of effects investigated (cf Box and Meyer 1986)

Effect hierarchy

• lower-order effects are more likely to be important than higher-order effects
• effects of the same order are equally likely to be important

Effect heredity

• interactions where at least one parent main effect is important are more likely to be important themselves

Wu and Hamada (2009), pp.172–172

Factorial designs can require a large number of runs for only a moderate number of factors (\(2^5 = 32\))

Resource constraints (eg cost) may mean not all \(2^m\) combinations can be run

Lots of degrees of freedom are devoted to estimating higher-order interactions

• eg in a \(2^5\) experiment, 16 degrees of freedom are used to estimate three-factor and higher-order interactions
• principles of effect hierarchy and sparsity suggest may be wasteful

Need to trade-off what you want to estimate against the number of runs you can afford

Production of bacteriocin (Morris 2011, p231)

• bacteriocin is a natural food preservative frown from bacteria

Unit

• a single bio-reaction

Treatment: combination of settings of the factors

• A: amount of glucose (\(x_1\))
• B: initial inoculum size (\(x_2\))
• C: level of aeration (\(x_3\))
• D: temperature (\(x_4\))
• E: amount of sodium (\(x_5\))

Assume each factor has two-levels, coded -1 and +1

Find an \(n=8\) run design using FrF2

bact.design <- FrF2(8, 5, factor.names = paste0("x", 1:5), 
     generators = list(c(1, 3), c(2, 3)), randomize = F, alias.info = 3)
bact.design

##   x1 x2 x3 x4 x5
## 1 -1 -1 -1  1  1
## 2  1 -1 -1 -1  1
## 3 -1  1 -1  1 -1
## 4  1  1 -1 -1 -1
## 5 -1 -1  1 -1 -1
## 6  1 -1  1  1 -1
## 7 -1  1  1 -1  1
## 8  1  1  1  1  1
## class=design, type= FrF2.generators

• \(8\) = \(32/4\) = \(2^5/2^2\) = \(2^{5-2}\)
• we need a principled way of choosing one-quarter of the runs from the factorial design that leads to clarity in the analysis

Assuming the number of runs is a power of two, \(n = 2^{k-q}\), we can construct \(2^{k-q} -1\) orthogonal vectors (with inner product zero), spanned by \(k-q = \log_2(n)\) vectors

• construct the full factorial design for \(k-q\) factors
• assign the remaining \(q\) factors to interaction columns

model.matrix(~ (x1 + x2 + x3) ^ 3, bact.design[, 1:3])[, ]

##   (Intercept) x11 x21 x31 x11:x21 x11:x31 x21:x31 x11:x21:x31
## 1           1  -1  -1  -1       1       1       1          -1
## 2           1   1  -1  -1      -1      -1       1           1
## 3           1  -1   1  -1      -1       1      -1           1
## 4           1   1   1  -1       1      -1      -1          -1
## 5           1  -1  -1   1       1      -1      -1           1
## 6           1   1  -1   1      -1       1      -1          -1
## 7           1  -1   1   1      -1      -1       1          -1
## 8           1   1   1   1       1       1       1           1

The design has been deliberately chosen so that

• \(x_4 = x_1x_3\)
• \(x_5 = x_2x_3\)

[\(x_1x_2\) is shorthand for the Hadamard (Schur or entry wise) product of two vectors, \(x_1\circ x_2\)]

What other consequences are there?

• \(x_4x_5 = x_1x_3x_2x_3 = x_1x_2x_3^2\)
• the product of any column with itself is the constant column (the identity)
• hence, \(x_4x_5 = x_1x_2\)

Now we can obtain the defining relation \(\ldots\)

• \(I = x_1x_3x_4 = x_2x_3x_5 = x_1x_2x_4x_5\)

\(\ldots\) and the complete aliasing scheme

• \(x_1 = x_3x_4 = x_1x_2x_3x_5 = x_2x_4x_5\)
• \(x_2 = x_1x_2x_3x_4 = x_3x_5 = x_1x_4x_5\)
• \(x_3 = x_1x_4 = x_2x_5 = x_1x_2x_3x_4x_5\)
• \(x_4 = x_1x_3 = x_2x_3x_4x_5 = x_1x_2x_5\)
• \(x_5 = x_1x_3x_4x_5 = x_2x_3 = x_1x_2x_4\)
• \(x_1x_2 = x_2x_3x_4 = x_1x_3x_5 = x_4x_5\)
• \(x_1x_5 = x_3x_4x_5 = x_1x_2x_3 = x_2x_4\)

FrF2 will summarise the aliasing amongst main effects and two- and three-factor interactions.

design.info(bact.design)$aliased 

## $legend
## [1] "A=x1" "B=x2" "C=x3" "D=x4" "E=x5"
## 
## $main
## [1] "A=CD=BDE" "B=CE=ADE" "C=AD=BE"  "D=AC=ABE" "E=BC=ABD"
## 
## $fi2
## [1] "AB=DE=ACE=BCD" "AE=BD=ABC=CDE"
## 
## $fi3
## [1] "ACD=BCE"

What is the consequence of this aliasing?

If more than one effect in each alias string is non-zero, the least squares estimators will be biased

• assumed model \(\boldsymbol{Y}= X_1\boldsymbol{\beta}_1 + \boldsymbol{\varepsilon}\)
• true model \(\boldsymbol{Y}= X_1\boldsymbol{\beta}_1 + X_2\boldsymbol{\beta}_2 + \boldsymbol{\varepsilon}\)

\[ \begin{split} E\left(\hat{\boldsymbol{\beta}}_1\right) & = \left(X_1^\mathrm{T}X_1\right)^{-1}X^\mathrm{T}_1E(\boldsymbol{Y}) \\ & = \left(X^\mathrm{T}_1X_1\right)^{-1}X_1^\mathrm{T}\left(X_1\boldsymbol{\beta}_1 + X_2\boldsymbol{\beta}_2\right) \\ & = \beta_1 + \left(X_1^\mathrm{T}X_1\right)^{-1}X_1^\mathrm{T}X_2\boldsymbol{\beta}_2 \\ & = \boldsymbol{\beta}_1 + A\boldsymbol{\beta}_2\\ \end{split} \]

\(A\) is the alias matrix

• if the columns of \(X_1\) and \(X_2\) are not orthogonal, \(\hat{\boldsymbol{\beta}}_1\) is biased

For the \(2^{5-2}\) example:

• \(X_1\) is an \(8\times 6\) matrix with columns for the intercept and five linear terms (“main effects”)
• \(X_2\) is an \(8\times 10\) matrix with columns for the 10 product terms (“two-factor interactions”)

\[ A = \left( \begin{array}{cccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \end{array} \right) \] For a regular design, the matrix \(A\) will only have entries 0, \(\pm 1\) (no aliasing or complete aliasing)

The transpose of the alias matrix is provided by the alias function.

ff.alias <- alias(y ~ (.)^2, data = data.frame(bact.design, y = vector(length = 8)))
ff.alias$Complete

##         (Intercept) x11 x21 x31 x41 x51 x11:x21 x11:x51
## x21:x31 0           0   0   0   0   1   0       0      
## x21:x41 0           0   0   0   0   0   0       1      
## x21:x51 0           0   0   1   0   0   0       0      
## x31:x41 0           1   0   0   0   0   0       0      
## x31:x51 0           0   1   0   0   0   0       0      
## x41:x51 0           0   0   0   0   0   1       0      
## x11:x31 0           0   0   0   1   0   0       0      
## x11:x41 0           0   0   1   0   0   0       0

These linear dependencies can be seen if we attempt to fit a linear model

bact.lm <- lm(yB ~ (x1 + x2 + x3 + x4 + x5)^2, data = bact)
summary(bact.lm)

## 
## Call:
## lm.default(formula = yB ~ (x1 + x2 + x3 + x4 + x5)^2, data = bact)
## 
## Residuals:
## ALL 8 residuals are 0: no residual degrees of freedom!
## 
## Coefficients: (8 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  4.42625         NA      NA       NA
## x1           0.26625         NA      NA       NA
## x2           0.24625         NA      NA       NA
## x3          -0.22875         NA      NA       NA
## x4          -1.11875         NA      NA       NA
## x5          -0.66375         NA      NA       NA
## x1:x2       -0.05375         NA      NA       NA
## x1:x3             NA         NA      NA       NA
## x1:x4             NA         NA      NA       NA
## x1:x5       -0.13375         NA      NA       NA
## x2:x3             NA         NA      NA       NA
## x2:x4             NA         NA      NA       NA
## x2:x5             NA         NA      NA       NA
## x3:x4             NA         NA      NA       NA
## x3:x5             NA         NA      NA       NA
## x4:x5             NA         NA      NA       NA
## 
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:    NaN 
## F-statistic:   NaN on 7 and 0 DF,  p-value: NA

Typically, in a first experiment, fractional factorial designs are used in screening

• investigate which of many factors have a substantive effect on the response
• main effects and two-factor interactions
• centre points to check for curvature

At second and later stages, augment the design

• to resolve ambiguities due to the aliasing of factorial effects (“break the alias strings”)
• to allow estimation of curvature and prediction from a more complex model

Regular fractional factorial designs have the number of runs equal to a power of the number of levels

• eg \(2^{5-2}\), \(3^{3-1}\times 2\)
• this inflexibility in run sizes can be a problem in practical experiments

Non-regular designs can have any number of runs (usually with \(n>p\), the number of parameters to be estimated)

Often the clarity provided by a regular design is lost

• no defining relation or straightforward aliasing scheme
• partial aliasing and fractional entries in \(A\)

One approach to finding non-regular designs is via a design optimality criterion

Notation: let \(\xi = [\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n]\) denote a design (choice of treatments and their replications)

Assuming the model \(\boldsymbol{Y}= X\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\), with \(\boldsymbol{\varepsilon}\sim N(0, \sigma^2I_n)\), a \(D\)-optimal design maximises \[ \phi(\xi) = \mathrm{det}\left(X^\mathrm{T}X\right) \]

That is, a \(D\)-optimal design maximises the determinant of the (expected) Fisher information matrix

• equivalent to minimising the volume of the joint confidence ellipsoid for \(\boldsymbol{\beta}\)

Also useful to define a Bayesian version, with \(R\) a prior precision matrix \[ \phi_B(\xi) = \mathrm{det}\left(X^\mathrm{T}X + R\right) \] (See later)

\(D\)-optimal designs are model dependent

• if the model (ie the columns of \(X\)) changes, the optimal design may change
• model-robust design is an active area of research

\(D\)-optimality promotes orthogonality in the \(X\) matrix

• if there are sufficient runs, the \(D\)-optimal design will usually be orthogonal
• for particular models and choices of \(n\), regular fractional factorial designs are \(D\)-optimal

There are many other optimality criteria, tailored to other experimental goals

• prediction, model discrimination, space-filling, …

\(k=11\) factors in \(n=12\) runs, first-order (main effects) model (Plackett and Burman 1946)

A particular \(D\)-optimal design is the following orthogonal array

Using the pb function in the FrF2 package:

pb.design <- pb(12, factor.names = paste0("x", 1:11))
pb.design

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11
## 1   1  1  1 -1 -1 -1  1 -1  1   1  -1
## 2  -1  1 -1  1  1 -1  1  1  1  -1  -1
## 3   1 -1 -1 -1  1 -1  1  1 -1   1   1
## 4   1  1 -1  1  1  1 -1 -1 -1   1  -1
## 5  -1  1  1 -1  1  1  1 -1 -1  -1   1
## 6  -1 -1  1 -1  1  1 -1  1  1   1  -1
## 7   1 -1  1  1 -1  1  1  1 -1  -1  -1
## 8   1  1 -1 -1 -1  1 -1  1  1  -1   1
## 9  -1 -1 -1  1 -1  1  1 -1  1   1   1
## 10  1 -1  1  1  1 -1 -1 -1  1  -1   1
## 11 -1 -1 -1 -1 -1 -1 -1 -1 -1  -1  -1
## 12 -1  1  1  1 -1 -1 -1  1 -1   1   1
## class=design, type= pb

This 12-run PB design is probably the most studied non-regular design

• orthogonal columns
• complex aliasing between main effects and two-factor interactions

pb.alias <- alias(y ~ (.)^2, data = data.frame(pb.design, y = vector(length = 12)))
head(pb.alias$Complete, n = 15)

##          (Intercept) x11  x21  x31  x41  x51  x61  x71  x81  x91  x101 x111
## x11:x21     0           0    0 -1/3 -1/3 -1/3  1/3 -1/3 -1/3  1/3  1/3 -1/3
## x11:x31     0           0 -1/3    0  1/3 -1/3 -1/3  1/3 -1/3  1/3 -1/3 -1/3
## x11:x41     0           0 -1/3  1/3    0  1/3  1/3 -1/3 -1/3 -1/3 -1/3 -1/3
## x11:x51     0           0 -1/3 -1/3  1/3    0 -1/3 -1/3 -1/3 -1/3  1/3  1/3
## x11:x61     0           0  1/3 -1/3  1/3 -1/3    0 -1/3  1/3 -1/3 -1/3 -1/3
## x11:x71     0           0 -1/3  1/3 -1/3 -1/3 -1/3    0  1/3 -1/3  1/3 -1/3
## x11:x81     0           0 -1/3 -1/3 -1/3 -1/3  1/3  1/3    0 -1/3 -1/3  1/3
## x11:x91     0           0  1/3  1/3 -1/3 -1/3 -1/3 -1/3 -1/3    0 -1/3  1/3
## x11:x101    0           0  1/3 -1/3 -1/3  1/3 -1/3  1/3 -1/3 -1/3    0 -1/3
## x11:x111    0           0 -1/3 -1/3 -1/3  1/3 -1/3 -1/3  1/3  1/3 -1/3    0
## x21:x31     0        -1/3    0    0 -1/3 -1/3 -1/3  1/3 -1/3 -1/3  1/3  1/3
## x21:x41     0        -1/3    0 -1/3    0  1/3 -1/3 -1/3  1/3 -1/3  1/3 -1/3
## x21:x51     0        -1/3    0 -1/3  1/3    0  1/3  1/3 -1/3 -1/3 -1/3 -1/3
## x21:x61     0         1/3    0 -1/3 -1/3  1/3    0 -1/3 -1/3 -1/3 -1/3  1/3
## x21:x71     0        -1/3    0  1/3 -1/3  1/3 -1/3    0 -1/3  1/3 -1/3 -1/3

Screening designs with fewer runs than factors (see Woods and Lewis 2017)

• can’t use ordinary least squares/maximum likelihood as \(X\) does not have full column rank
• Bayesian \(D\)-optimality with \(R = [0\,|\, \tau I_m]\)

Supersaturated experiment used by GlaxoSmithKline in the development of a new oncology drug

• \(k=16\) factors: e.g. temperature, solvent amount, reaction time
• \(n=10\) runs
• Bayesian \(D\)-optimal design with \(\tau = 0.2\)

ssd

##    x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16
## 1   1  1 -1  1  1 -1 -1 -1 -1  -1  -1   1   1   1   1   1
## 2   1  1  1 -1 -1 -1 -1 -1  1   1  -1  -1   1  -1  -1   1
## 3  -1 -1  1 -1  1 -1  1 -1 -1   1  -1  -1  -1   1   1  -1
## 4  -1  1  1  1  1 -1 -1  1 -1  -1   1  -1  -1  -1  -1  -1
## 5  -1 -1 -1 -1 -1  1 -1  1  1  -1  -1  -1  -1  -1   1   1
## 6   1  1  1 -1  1  1 -1  1  1   1   1   1   1   1   1  -1
## 7  -1 -1  1  1 -1 -1  1 -1  1  -1   1   1   1  -1   1  -1
## 8   1 -1 -1  1  1  1  1  1 -1   1  -1   1   1  -1  -1  -1
## 9  -1  1 -1  1 -1  1  1 -1  1   1   1  -1  -1   1  -1   1
## 10  1 -1  1 -1 -1 -1  1  1 -1  -1   1   1  -1   1  -1   1

Partial aliasing between main effects

Heatmap of column correlations:

library(fields)
par(mar=c(8,2,0,0))
image.plot(1:16,1:16, cor(ssd), zlim = c(-1, 1), xlab = "Factors", 
           ylab = "", asp = 1, axes = F)
axis(1, at = seq(2, 16, by = 2), line = .5)
axis(2, at = seq(2, 16, by = 2), line = -5)

Analysis via regularised (shrinkage) methods (eg lasso, Dantzig selector; see APTS High Dimensional Statistics)

• small coefficients shrunk to zero

Now consider a more general class of models (cf preliminary material).

Let \(\boldsymbol{y}= (y_1,\ldots,y_n)^\mathrm{T}\) be iid observations from a distribution with density/mass function \(\pi(y_i\,;\,\boldsymbol{\theta},\boldsymbol{x}_i)\)

• \(\boldsymbol{\theta}\) is a \(q-\)vector of unknown parameters
• \(\boldsymbol{x}_i =(x_{1i},\ldots,x_{ki})^\mathrm{T}\) is a vector of values of \(k\) controllable variables.

The (expected) information matrix \[ M(\boldsymbol{\theta}) = E_y\left[-\frac{\partial^2l(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^\mathrm{T}}\right] \] is an important quantity for design, where \(l(\boldsymbol{\theta}) = \sum_{i=1}^n\log\pi(y_i;\,\boldsymbol{\theta},\boldsymbol{x}_i)\) (the log-likelihood).

• \(M(\boldsymbol{\theta})\) is the (asymptotic) precision for the maximum likelihood estimators \(\hat{\boldsymbol{\theta}}\).
• \(M(\boldsymbol{\theta})\) is also an asymptotic approximation to the posterior precision for \(\boldsymbol{\theta}\) in a Bayesian analysis.

Example 1: Compartmental model \[ y_i \sim N\left(c(\boldsymbol{\theta})\mu(\boldsymbol{\theta};\,x_i), \sigma^2\nu(\boldsymbol{\theta};\,x_i)\right)\,,\quad x_i\in[0,24]\,, \] with \[ \mu(\boldsymbol{\theta};\,x) = \exp(-\theta_1x)-exp(-\theta_2x)\,,\quad c(\boldsymbol{\theta}) = \frac{400\theta_2}{\theta_3(\theta_2-\theta_1)}\,,\quad \nu(\boldsymbol{\theta};\,x) = 1 + \frac{\tau^2}{\sigma^2}c(\boldsymbol{\theta})^2\mu(\boldsymbol{\theta};\,x)\,, \] for \(\theta_1, \theta_2, \theta_3, \tau^2, \sigma^2>0\).

Prior distributions (for later use):

• \(\log\theta_i\sim N(m_i, 0.05)\), with \(m_1 = \log 0.1, m_2 = 0, m_3 = \log 20\)

Ryan et al. (2014)

comp <- function(x, theta, D = 400) {
    mu <- exp(-theta[1] * x) - exp(-theta[2] * x)
    c <- (D / theta[3]) * (theta[2]) / (theta[2] - theta[1])
    c * mu }
theta <- c(.1, 1, 20)
M <- 100
par(mar = c(6, 4, 0, 1) + .1)
lapply(1:M, function(l) {
  thetat <- rlnorm(3,log(theta),rep(0.05,3))
  curve(comp(x, theta = thetat), from = 0, to = 24, ylab = "Expected concentration", 
        xlab = "Time", ylim = c(0, 20), xlim = c(0, 24), add = l!=1) })

## [[1]]
## [[1]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[1]]$y
##   [1]  0.000000  4.387558  7.706913 10.194512 12.034989 13.372503 14.319597
##   [8] 14.964126 15.374653 15.604680 15.695942 15.680984 15.585177 15.428283
##  [15] 15.225691 14.989372 14.728627 14.450675 14.161109 13.864252 13.563440
##  [22] 13.261231 12.959587 12.659995 12.363581 12.071183 11.783420 11.500736
##  [29] 11.223444 10.951753 10.685793 10.425630 10.171284  9.922741  9.679956
##  [36]  9.442865  9.211390  8.985438  8.764912  8.549707  8.339714  8.134823
##  [43]  7.934922  7.739899  7.549643  7.364043  7.182989  7.006374  6.834092
##  [50]  6.666039  6.502111  6.342211  6.186239  6.034100  5.885700  5.740948
##  [57]  5.599754  5.462033  5.327697  5.196665  5.068855  4.944188  4.822586
##  [64]  4.703976  4.588282  4.475434  4.365361  4.257995  4.153270  4.051120
##  [71]  3.951483  3.854296  3.759500  3.667035  3.576844  3.488871  3.403062
##  [78]  3.319364  3.237724  3.158092  3.080419  3.004656  2.930756  2.858674
##  [85]  2.788365  2.719785  2.652892  2.587644  2.524001  2.461923  2.401372
##  [92]  2.342310  2.284701  2.228508  2.173698  2.120236  2.068089  2.017224
##  [99]  1.967610  1.919217  1.872014
## 
## 
## [[2]]
## [[2]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[2]]$y
##   [1]  0.000000  4.418766  7.765400 10.275523 12.133537 13.483741 14.439039
##   [8] 15.087758 15.498986 15.726747 15.813275 15.791574 15.687424 15.520950
##  [15] 15.307852 15.060365 14.788015 14.498208 14.196690 13.887910 13.575302
##  [22] 13.261508 12.948550 12.637965 12.330914 12.028262 11.730645 11.438520
##  [29] 11.152205 10.871911 10.597765 10.329831 10.068121  9.812612  9.563251
##  [36]  9.319963  9.082659  8.851235  8.625582  8.405582  8.191116  7.982062
##  [43]  7.778296  7.579694  7.386134  7.197494  7.013654  6.834496  6.659903
##  [50]  6.489762  6.323961  6.162391  6.004944  5.851517  5.702007  5.556316
##  [57]  5.414345  5.276001  5.141191  5.009824  4.881814  4.757074  4.635521
##  [64]  4.517074  4.401653  4.289181  4.179583  4.072786  3.968717  3.867307
##  [71]  3.768488  3.672195  3.578362  3.486927  3.397828  3.311005  3.226401
##  [78]  3.143959  3.063624  2.985341  2.909059  2.834725  2.762292  2.691709
##  [85]  2.622929  2.555907  2.490598  2.426957  2.364943  2.304513  2.245627
##  [92]  2.188246  2.132332  2.077846  2.024752  1.973015  1.922600  1.873473
##  [99]  1.825601  1.778953  1.733496
## 
## 
## [[3]]
## [[3]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[3]]$y
##   [1]  0.000000  4.444527  7.798034 10.307158 12.163191 13.514493 14.476116
##   [8] 15.137278 15.567150 15.819359 15.935466 15.947680 15.880951 15.754598
##  [15] 15.583576 15.379452 15.151167 14.905625 14.648149 14.382840 14.112848
##  [22] 13.840586 13.567898 13.296187 13.026512 12.759668 12.496246 12.236677
##  [29] 11.981271 11.730243 11.483736 11.241836 11.004587 10.772000 10.544062
##  [36] 10.320739 10.101986  9.887744  9.677950  9.472531  9.271414  9.074522
##  [43]  8.881777  8.693098  8.508406  8.327622  8.150666  7.977460  7.807928
##  [50]  7.641993  7.479579  7.320614  7.165024  7.012739  6.863689  6.717806
##  [57]  6.575022  6.435272  6.298492  6.164619  6.033591  5.905348  5.779830
##  [64]  5.656980  5.536741  5.419057  5.303875  5.191141  5.080803  4.972810
##  [71]  4.867113  4.763662  4.662410  4.563310  4.466317  4.371385  4.278471
##  [78]  4.187532  4.098525  4.011411  3.926148  3.842697  3.761021  3.681080
##  [85]  3.602838  3.526260  3.451309  3.377951  3.306152  3.235880  3.167101
##  [92]  3.099784  3.033898  2.969412  2.906297  2.844523  2.784063  2.724887
##  [99]  2.666969  2.610283  2.554801
## 
## 
## [[4]]
## [[4]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[4]]$y
##   [1]  0.000000  4.216964  7.395974  9.770283 11.521201 12.789655 13.685169
##   [8] 14.292850 14.678825 14.894463 14.979653 14.965369 14.875645 14.729126
##  [15] 14.540263 14.320251 14.077753 13.819464 13.550547 13.274981 12.995819
##  [22] 12.715399 12.435502 12.157477 11.882339 11.610843 11.343542 11.080835
##  [29] 10.823000 10.570223 10.322617 10.080242  9.843116  9.611224  9.384528
##  [36]  9.162972  8.946484  8.734985  8.528389  8.326603  8.129532  7.937080
##  [43]  7.749148  7.565638  7.386452  7.211493  7.040666  6.873874  6.711027
##  [50]  6.552031  6.396797  6.245237  6.097265  5.952797  5.811751  5.674045
##  [57]  5.539600  5.408341  5.280191  5.155077  5.032927  4.913672  4.797242
##  [64]  4.683570  4.572592  4.464243  4.358462  4.255187  4.154359  4.055920
##  [71]  3.959814  3.865985  3.774379  3.684944  3.597628  3.512381  3.429154
##  [78]  3.347899  3.268570  3.191120  3.115505  3.041682  2.969609  2.899243
##  [85]  2.830544  2.763474  2.697992  2.634062  2.571647  2.510711  2.451219
##  [92]  2.393137  2.336431  2.281068  2.227018  2.174248  2.122728  2.072429
##  [99]  2.023322  1.975379  1.928572
## 
## 
## [[5]]
## [[5]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[5]]$y
##   [1]  0.000000  4.218973  7.397464  9.769184 11.515852 12.778748 13.667702
##   [8] 14.268083 14.646235 14.853707 14.930539 14.907824 14.809696 14.654875
##  [15] 14.457877 14.229942 13.979770 13.714081 13.438060 13.155696 12.870052
##  [22] 12.583468 12.297726 12.014174 11.733822 11.457419 11.185513 10.918495
##  [29] 10.656634 10.400107 10.149021  9.903424  9.663327  9.428704  9.199509
##  [36]  8.975677  8.757127  8.543773  8.335519  8.132264  7.933906  7.740340
##  [43]  7.551461  7.367162  7.187340  7.011890  6.840710  6.673698  6.510756
##  [50]  6.351786  6.196693  6.045383  5.897764  5.753749  5.613247  5.476176
##  [57]  5.342451  5.211990  5.084714  4.960546  4.839410  4.721232  4.605939
##  [64]  4.493461  4.383731  4.276679  4.172242  4.070355  3.970956  3.873985
##  [71]  3.779381  3.687088  3.597048  3.509207  3.423512  3.339909  3.258347
##  [78]  3.178778  3.101151  3.025420  2.951539  2.879462  2.809144  2.740544
##  [85]  2.673620  2.608329  2.544633  2.482493  2.421869  2.362727  2.305028
##  [92]  2.248739  2.193824  2.140251  2.087985  2.036996  1.987252  1.938723
##  [99]  1.891379  1.845191  1.800131
## 
## 
## [[6]]
## [[6]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[6]]$y
##   [1]  0.000000  3.739786  6.660206  8.923394 10.659772 11.974240 12.951165
##   [8] 13.658391 14.150467 14.471243 14.655963 14.732945 14.724935 14.650197
##  [15] 14.523387 14.356262 14.158243 13.936877 13.698199 13.447031 13.187219
##  [22] 12.921824 12.653278 12.383503 12.114019 11.846014 11.580418 11.317948
##  [29] 11.059155 10.804453 10.554147 10.308460 10.067543  9.831492  9.600361
##  [36]  9.374170  9.152911  8.936554  8.725054  8.518353  8.316382  8.119066
##  [43]  7.926323  7.738069  7.554217  7.374677  7.199358  7.028171  6.861025
##  [50]  6.697831  6.538499  6.382943  6.231075  6.082810  5.938066  5.796759
##  [57]  5.658810  5.524140  5.392671  5.264328  5.139038  5.016727  4.897327
##  [64]  4.780767  4.666980  4.555901  4.447465  4.341610  4.238273  4.137396
##  [71]  4.038920  3.942787  3.848942  3.757331  3.667900  3.580598  3.495374
##  [78]  3.412178  3.330962  3.251679  3.174284  3.098730  3.024975  2.952975
##  [85]  2.882689  2.814076  2.747096  2.681710  2.617880  2.555570  2.494743
##  [92]  2.435364  2.377398  2.320811  2.265572  2.211647  2.159006  2.107618
##  [99]  2.057452  2.008481  1.960676
## 
## 
## [[7]]
## [[7]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[7]]$y
##   [1]  0.000000  4.505233  7.949238 10.559416 12.514918 13.956877 14.996461
##   [8] 15.721224 16.200114 16.487416 16.625860 16.649069 16.583494 16.449933
##  [15] 16.264728 16.040714 15.787960 15.514359 15.226088 14.927974 14.623779
##  [22] 14.316431 14.008196 13.700823 13.395655 13.093711 12.795763 12.502381
##  [29] 12.213983 11.930866 11.653231 11.381205 11.114858 10.854216 10.599270
##  [36] 10.349984 10.106302  9.868154  9.635458  9.408122  9.186050  8.969141
##  [43]  8.757292  8.550398  8.348354  8.151054  7.958393  7.770266  7.586572
##  [50]  7.407209  7.232077  7.061079  6.894118  6.731101  6.571934  6.416529
##  [57]  6.264796  6.116650  5.972005  5.830780  5.692893  5.558267  5.426824
##  [64]  5.298489  5.173188  5.050851  4.931406  4.814785  4.700923  4.589753
##  [71]  4.481212  4.375238  4.271770  4.170748  4.072116  3.975816  3.881794
##  [78]  3.789995  3.700367  3.612858  3.527419  3.444000  3.362554  3.283035
##  [85]  3.205395  3.129592  3.055582  2.983321  2.912770  2.843887  2.776633
##  [92]  2.710969  2.646859  2.584264  2.523150  2.463481  2.405223  2.348343
##  [99]  2.292808  2.238586  2.185646
## 
## 
## [[8]]
## [[8]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[8]]$y
##   [1]  0.000000  4.012339  7.059673  9.353897 11.060801 12.310092 13.203237
##   [8] 13.819604 14.221265 14.456756 14.564021 14.572713 14.506000 14.381974
##  [15] 14.214756 14.015357 13.792360 13.552443 13.300798 13.041455 12.777531
##  [22] 12.511433 12.245013 11.979688 11.716535 11.456369 11.199794 10.947258
##  [29] 10.699080 10.455483 10.216616  9.982566  9.753377  9.529059  9.309594
##  [36]  9.094944  8.885055  8.679863  8.479295  8.283272  8.091710  7.904523
##  [43]  7.721623  7.542921  7.368329  7.197757  7.031118  6.868324  6.709290
##  [50]  6.553930  6.402161  6.253902  6.109073  5.967595  5.829391  5.694386
##  [57]  5.562506  5.433680  5.307836  5.184906  5.064822  4.947520  4.832933
##  [64]  4.721001  4.611660  4.504852  4.400518  4.298599  4.199041  4.101789
##  [71]  4.006790  3.913990  3.823340  3.734789  3.648289  3.563793  3.481253
##  [78]  3.400625  3.321865  3.244928  3.169774  3.096360  3.024646  2.954594
##  [85]  2.886164  2.819318  2.754021  2.690236  2.627929  2.567065  2.507610
##  [92]  2.449532  2.392799  2.337381  2.283246  2.230364  2.178708  2.128247
##  [99]  2.078956  2.030806  1.983771
## 
## 
## [[9]]
## [[9]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[9]]$y
##   [1]  0.000000  4.299219  7.563820 10.020767 11.847707 13.183668 14.137451
##   [8] 14.794186 15.220474 15.468408 15.578727 15.583279 15.506952 15.369187
##  [15] 15.185159 14.966707 14.723058 14.461395 14.187301 13.905106 13.618165
##  [22] 13.329064 13.039794 12.751874 12.466463 12.184430 11.906424 11.632921
##  [29] 11.364261 11.100678 10.842328 10.589300 10.341637 10.099343  9.862396
##  [36]  9.630748  9.404339  9.183094  8.966929  8.755755  8.549479  8.348002
##  [43]  8.151227  7.959053  7.771382  7.588113  7.409149  7.234392  7.063746
##  [50]  6.897117  6.734412  6.575540  6.420412  6.268940  6.121040  5.976627
##  [57]  5.835619  5.697937  5.563503  5.432240  5.304073  5.178929  5.056738
##  [64]  4.937430  4.820936  4.707191  4.596129  4.487688  4.381805  4.278420
##  [71]  4.177474  4.078910  3.982672  3.888704  3.796954  3.707368  3.619896
##  [78]  3.534487  3.451094  3.369668  3.290164  3.212535  3.136738  3.062729
##  [85]  2.990467  2.919909  2.851016  2.783749  2.718069  2.653938  2.591321
##  [92]  2.530181  2.470483  2.412194  2.355280  2.299709  2.245450  2.192470
##  [99]  2.140741  2.090232  2.040914
## 
## 
## [[10]]
## [[10]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[10]]$y
##   [1]  0.000000  4.223766  7.469141  9.943156 11.809427 13.197250 14.208791
##   [8] 14.924772 15.408960 15.711727 15.872848 15.923730 15.889160 15.788697
##  [15] 15.637764 15.448519 15.230536 14.991351 14.736886 14.471792 14.199710
##  [22] 13.923491 13.645354 13.367027 13.089844 12.814832 12.542774 12.274262
##  [29] 12.009738 11.749524 11.493850 11.242870 10.996685 10.755349 10.518880
##  [36] 10.287272 10.060496  9.838507  9.621250  9.408658  9.200659  8.997175
##  [43]  8.798127  8.603431  8.413002  8.226756  8.044607  7.866472  7.692265
##  [50]  7.521903  7.355305  7.192388  7.033074  6.877284  6.724940  6.575969
##  [57]  6.430295  6.287846  6.148551  6.012341  5.879147  5.748903  5.621544
##  [64]  5.497006  5.375227  5.256145  5.139701  5.025837  4.914495  4.805619
##  [71]  4.699156  4.595051  4.493252  4.393708  4.296370  4.201188  4.108115
##  [78]  4.017104  3.928109  3.841085  3.755990  3.672779  3.591412  3.511848
##  [85]  3.434047  3.357968  3.283576  3.210831  3.139699  3.070141  3.002125
##  [92]  2.935616  2.870580  2.806985  2.744799  2.683991  2.624530  2.566386
##  [99]  2.509530  2.453934  2.399569
## 
## 
## [[11]]
## [[11]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[11]]$y
##   [1]  0.000000  4.258450  7.462424  9.850480 11.607687 12.877608 13.771599
##   [8] 14.376022 14.757846 14.968990 15.049695 15.031142 14.937482 14.787411
##  [15] 14.595394 14.372612 14.127700 13.867318 13.596593 13.319466 13.038955
##  [22] 12.757366 12.476452 12.197536 11.921612 11.649415 11.381485 11.118205
##  [29] 10.859845 10.606578 10.358514 10.115704  9.878161  9.645867  9.418780
##  [36]  9.196842  8.979979  8.768110  8.561148  8.358999  8.161567  7.968755
##  [43]  7.780465  7.596597  7.417055  7.241740  7.070557  6.903411  6.740209
##  [50]  6.580859  6.425273  6.273361  6.125038  5.980220  5.838825  5.700771
##  [57]  5.565981  5.434377  5.305884  5.180428  5.057939  4.938345  4.821580
##  [64]  4.707574  4.596265  4.487587  4.381479  4.277879  4.176729  4.077971
##  [71]  3.981548  3.887405  3.795488  3.705744  3.618122  3.532572  3.449044
##  [78]  3.367492  3.287868  3.210127  3.134224  3.060115  2.987759  2.917114
##  [85]  2.848139  2.780795  2.715043  2.650846  2.588167  2.526970  2.467220
##  [92]  2.408883  2.351925  2.296314  2.242018  2.189006  2.137247  2.086712
##  [99]  2.037372  1.989199  1.942164
## 
## 
## [[12]]
## [[12]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[12]]$y
##   [1]  0.000000  4.025308  7.106623  9.445348 11.200338 12.496880 13.433778
##   [8] 14.088937 14.523779 14.786721 14.915918 14.941430 14.886930 14.771057
##  [15] 14.608474 14.410709 14.186821 13.943918 13.687569 13.422134 13.151015
##  [22] 12.876864 12.601736 12.327221 12.054541 11.784629 11.518187 11.255743
##  [29] 10.997681 10.744275 10.495715 10.252122 10.013564  9.780071  9.551637
##  [36]  9.328235  9.109820  8.896330  8.687694  8.483833  8.284664  8.090097
##  [43]  7.900042  7.714406  7.533096  7.356020  7.183083  7.014195  6.849263
##  [50]  6.688199  6.530913  6.377320  6.227333  6.080870  5.937848  5.798187
##  [57]  5.661809  5.528637  5.398596  5.271613  5.147616  5.026535  4.908301
##  [64]  4.792848  4.680111  4.570025  4.462528  4.357560  4.255060  4.154972
##  [71]  4.057238  3.961802  3.868612  3.777613  3.688755  3.601987  3.517260
##  [78]  3.434526  3.353738  3.274850  3.197818  3.122598  3.049148  2.977425
##  [85]  2.907389  2.839000  2.772220  2.707011  2.643336  2.581159  2.520444
##  [92]  2.461157  2.403265  2.346735  2.291534  2.237632  2.184998  2.133601
##  [99]  2.083414  2.034407  1.986554
## 
## 
## [[13]]
## [[13]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[13]]$y
##   [1]  0.000000  4.233650  7.487927  9.968033 11.836641 13.222698 14.228406
##   [8] 14.934760 15.405935 15.692774 15.835540 15.866117 15.809736 15.686359
##  [15] 15.511769 15.298435 15.056202 14.792829 14.514430 14.225809 13.930735
##  [22] 13.632158 13.332375 13.033172 12.735927 12.441697 12.151283 11.865288
##  [29] 11.584153 11.308198 11.037642 10.772628 10.513240 10.259512 10.011444
##  [36]  9.769007  9.532149  9.300802  9.074887  8.854314  8.638988  8.428806
##  [43]  8.223665  8.023460  7.828083  7.637427  7.451387  7.269855  7.092728
##  [50]  6.919902  6.751276  6.586750  6.426226  6.269609  6.116805  5.967721
##  [57]  5.822267  5.680357  5.541904  5.406824  5.275035  5.146458  5.021014
##  [64]  4.898628  4.779224  4.662730  4.549075  4.438191  4.330009  4.224464
##  [71]  4.121491  4.021029  3.923015  3.827390  3.734096  3.643077  3.554275
##  [78]  3.467639  3.383114  3.300649  3.220194  3.141701  3.065121  2.990407
##  [85]  2.917515  2.846400  2.777017  2.709327  2.643286  2.578855  2.515994
##  [92]  2.454666  2.394832  2.336457  2.279505  2.223942  2.169732  2.116844
##  [99]  2.065245  2.014904  1.965790
## 
## 
## [[14]]
## [[14]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[14]]$y
##   [1]  0.000000  4.383493  7.693563 10.169706 11.998490 13.325243 14.263155
##   [8] 14.900374 15.305534 15.532065 15.621550 15.606342 15.511600 15.356882
##  [15] 15.157380 14.924884 14.668539 14.395422 14.111007 13.819517 13.524199
##  [22] 13.227546 12.931457 12.637375 12.346388 12.059304 11.776719 11.499061
##  [29] 11.226629 10.959625 10.698169 10.442328 10.192119  9.947527  9.708509
##  [36]  9.475005  9.246938  9.024222  8.806760  8.594455  8.387201  8.184894
##  [43]  7.987427  7.794692  7.606584  7.422996  7.243825  7.068967  6.898320
##  [50]  6.731787  6.569268  6.410668  6.255894  6.104854  5.957459  5.813621
##  [57]  5.673255  5.536276  5.402604  5.272160  5.144864  5.020641  4.899418
##  [64]  4.781121  4.665680  4.553027  4.443093  4.335814  4.231125  4.128964
##  [71]  4.029269  3.931982  3.837043  3.744397  3.653988  3.565761  3.479665
##  [78]  3.395648  3.313659  3.233650  3.155573  3.079381  3.005028  2.932471
##  [85]  2.861666  2.792571  2.725143  2.659344  2.595134  2.532473  2.471326
##  [92]  2.411656  2.353426  2.296602  2.241150  2.187036  2.134230  2.082698
##  [99]  2.032411  1.983338  1.935450
## 
## 
## [[15]]
## [[15]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[15]]$y
##   [1]  0.000000  4.420181  7.770998 10.287385 12.153189 13.512270 14.477182
##   [8] 15.135975 15.557518 15.795664 15.892515 15.880974 15.786741 15.629886
##  [15] 15.426064 15.187482 14.923646 14.641948 14.348129 14.046637 13.740909
##  [22] 13.433590 13.126710 12.821817 12.520079 12.222374 11.929349 11.641473
##  [29] 11.359076 11.082382 10.811529 10.546593 10.287599 10.034535  9.787359
##  [36]  9.546008  9.310402  9.080449  8.856049  8.637096  8.423479  8.215084
##  [43]  8.011797  7.813504  7.620089  7.431440  7.247443  7.067987  6.892964
##  [50]  6.722267  6.555790  6.393431  6.235088  6.080664  5.930062  5.783188
##  [57]  5.639950  5.500259  5.364026  5.231168  5.101599  4.975239  4.852008
##  [64]  4.731830  4.614628  4.500329  4.388860  4.280153  4.174138  4.070749
##  [71]  3.969920  3.871589  3.775694  3.682174  3.590970  3.502025  3.415283
##  [78]  3.330690  3.248192  3.167737  3.089275  3.012757  2.938134  2.865359
##  [85]  2.794387  2.725172  2.657673  2.591845  2.527647  2.465040  2.403983
##  [92]  2.344439  2.286369  2.229738  2.174509  2.120649  2.068122  2.016897
##  [99]  1.966940  1.918221  1.870709
## 
## 
## [[16]]
## [[16]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[16]]$y
##   [1]  0.000000  3.730660  6.614252  8.824850 10.501147 11.753640 12.670351
##   [8] 13.321392 13.762598 14.038426 14.184268 14.228287 14.192889 14.095890
##  [15] 13.951451 13.770818 13.562916 13.334820 13.092134 12.839286 12.579771
##  [22] 12.316341 12.051155 11.785901 11.521892 11.260144 11.001435 10.746357
##  [29] 10.495352 10.248742 10.006759  9.769559  9.537240  9.309855  9.087421
##  [36]  8.869927  8.657339  8.449608  8.246672  8.048458  7.854888  7.665878
##  [43]  7.481339  7.301181  7.125314  6.953643  6.786078  6.622526  6.462896
##  [50]  6.307099  6.155044  6.006646  5.861817  5.720474  5.582534  5.447917
##  [57]  5.316542  5.188333  5.063214  4.941110  4.821950  4.705662  4.592178
##  [64]  4.481430  4.373352  4.267881  4.164953  4.064507  3.966483  3.870823
##  [71]  3.777470  3.686368  3.597463  3.510702  3.426034  3.343408  3.262774
##  [78]  3.184085  3.107294  3.032354  2.959222  2.887854  2.818207  2.750239
##  [85]  2.683911  2.619183  2.556015  2.494371  2.434213  2.375507  2.318216
##  [92]  2.262307  2.207746  2.154502  2.102541  2.051833  2.002349  1.954058
##  [99]  1.906931  1.860941  1.816060
## 
## 
## [[17]]
## [[17]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[17]]$y
##   [1]  0.000000  4.363251  7.696426 10.218752 12.103357 13.486946 14.477454
##   [8] 15.160083 15.602081 15.856508 15.965219 15.961218 15.870517 15.713607
##  [15] 15.506616 15.262229 14.990410 14.698977 14.394053 14.080422 13.761814
##  [22] 13.441127 13.120599 12.801953 12.486505 12.175250 11.868928 11.568085
##  [29] 11.273108 10.984264 10.701724 10.425586 10.155890  9.892630  9.635768
##  [36]  9.385238  9.140956  8.902822  8.670726  8.444549  8.224168  8.009457
##  [43]  7.800287  7.596528  7.398051  7.204729  7.016432  6.833037  6.654420
##  [50]  6.480460  6.311037  6.146036  5.985343  5.828846  5.676437  5.528011
##  [57]  5.383463  5.242693  5.105602  4.972095  4.842078  4.715460  4.592153
##  [64]  4.472070  4.355126  4.241240  4.130332  4.022324  3.917141  3.814707
##  [71]  3.714953  3.617806  3.523201  3.431069  3.341346  3.253969  3.168878
##  [78]  3.086011  3.005312  2.926722  2.850188  2.775656  2.703072  2.632386
##  [85]  2.563549  2.496512  2.431228  2.367651  2.305737  2.245441  2.186723
##  [92]  2.129540  2.073852  2.019620  1.966807  1.915375  1.865288  1.816510
##  [99]  1.769008  1.722748  1.677698
## 
## 
## [[18]]
## [[18]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[18]]$y
##   [1]  0.000000  3.714132  6.549903  8.695824 10.300367 11.480462 12.328187
##   [8] 12.916039 13.301091 13.528262 13.632898 13.642801 13.579835 13.461180
##  [15] 13.300333 13.107885 12.892142 12.659608 12.415371 12.163401 11.906791
##  [22] 11.647941 11.388709 11.130524 10.874479 10.621405 10.371923 10.126493
##  [29]  9.885447  9.649019  9.417363  9.190574  8.968699  8.751749  8.539708
##  [36]  8.332537  8.130181  7.932573  7.739637  7.551291  7.367447  7.188016
##  [43]  7.012903  6.842018  6.675264  6.512551  6.353784  6.198872  6.047726
##  [50]  5.900255  5.756372  5.615993  5.479032  5.345408  5.215039  5.087848
##  [57]  4.963758  4.842692  4.724578  4.609344  4.496920  4.387238  4.280230
##  [64]  4.175832  4.073980  3.974612  3.877667  3.783087  3.690814  3.600791
##  [71]  3.512964  3.427279  3.343684  3.262128  3.182562  3.104935  3.029203
##  [78]  2.955317  2.883234  2.812908  2.744299  2.677362  2.612058  2.548347
##  [85]  2.486190  2.425549  2.366388  2.308669  2.252358  2.197420  2.143823
##  [92]  2.091533  2.040518  1.990747  1.942191  1.894819  1.848602  1.803513
##  [99]  1.759523  1.716606  1.674736
## 
## 
## [[19]]
## [[19]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[19]]$y
##   [1]  0.000000  4.338871  7.620632 10.080246 11.900930 13.225548 14.165480
##   [8] 14.807554 15.219444 15.453891 15.551980 15.545714 15.460006 15.314245
##  [15] 15.123506 14.899508 14.651347 14.386076 14.109155 13.824805 13.536276
##  [22] 13.246067 12.956090 12.667799 12.382294 12.100395 11.822712 11.549684
##  [29] 11.281624 11.018742 10.761172 10.508989 10.262221 10.020862  9.784878
##  [36]  9.554218  9.328812  9.108581  8.893438  8.683290  8.478039  8.277588
##  [43]  8.081835  7.890678  7.704019  7.521755  7.343788  7.170020  7.000354
##  [50]  6.834696  6.672952  6.515032  6.360845  6.210305  6.063326  5.919823
##  [57]  5.779716  5.642923  5.509367  5.378972  5.251663  5.127366  5.006011
##  [64]  4.887527  4.771848  4.658907  4.548639  4.440980  4.335870  4.233247
##  [71]  4.133053  4.035231  3.939723  3.846477  3.755437  3.666552  3.579771
##  [78]  3.495043  3.412321  3.331557  3.252705  3.175719  3.100555  3.027170
##  [85]  2.955521  2.885569  2.817272  2.750592  2.685490  2.621929  2.559872
##  [92]  2.499284  2.440130  2.382376  2.325989  2.270937  2.217188  2.164711
##  [99]  2.113475  2.063453  2.014614
## 
## 
## [[20]]
## [[20]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[20]]$y
##   [1]  0.000000  4.653425  8.199243 10.878198 12.879182 14.350390 15.408076
##   [8] 16.143431 16.627974 16.917787 17.056836 17.079580 17.013019 16.878297
##  [15] 16.691964 16.466969 16.213429 15.939246 15.650580 15.352227 15.047913
##  [22] 14.740523 14.432286 14.124917 13.819724 13.517701 13.219595 12.925958
##  [29] 12.637194 12.353587 12.075329 11.802543 11.535294 11.273606 11.017470
##  [36] 10.766851 10.521697 10.281938 10.047497  9.818288  9.594217  9.375189
##  [43]  9.161106  8.951869  8.747376  8.547528  8.352225  8.161367  7.974858
##  [50]  7.792602  7.614502  7.440467  7.270404  7.104225  6.941841  6.783166
##  [57]  6.628117  6.476610  6.328565  6.183903  6.042547  5.904422  5.769454
##  [64]  5.637571  5.508702  5.382779  5.259734  5.139501  5.022017  4.907219
##  [71]  4.795044  4.685434  4.578329  4.473673  4.371409  4.271482  4.173840
##  [78]  4.078430  3.985200  3.894102  3.805087  3.718106  3.633113  3.550063
##  [85]  3.468912  3.389616  3.312132  3.236420  3.162438  3.090147  3.019509
##  [92]  2.950486  2.883040  2.817137  2.752739  2.689814  2.628327  2.568246
##  [99]  2.509538  2.452172  2.396118
## 
## 
## [[21]]
## [[21]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[21]]$y
##   [1]  0.000000  3.802871  6.691847  8.865725 10.480534 11.658721 12.496371
##   [8] 13.068868 13.435345 13.642177 13.725722 13.714475 13.630759 13.492052
##  [15] 13.312026 13.101371 12.868434 12.619723 12.360305 12.094115 11.824201
##  [22] 11.552916 11.282067 11.013036 10.746869 10.484351 10.226064  9.972427
##  [29]  9.723738  9.480197  9.241929  9.009001  8.781434  8.559215  8.342306
##  [36]  8.130647  7.924165  7.722775  7.526384  7.334894  7.148202  6.966204
##  [43]  6.788795  6.615868  6.447319  6.283041  6.122932  5.966889  5.814813
##  [50]  5.666604  5.522166  5.381405  5.244227  5.110544  4.980265  4.853306
##  [57]  4.729582  4.609010  4.491512  4.377008  4.265422  4.156681  4.050712
##  [64]  3.947444  3.846808  3.748738  3.653168  3.560034  3.469274  3.380829
##  [71]  3.294638  3.210644  3.128792  3.049026  2.971294  2.895544  2.821724
##  [78]  2.749787  2.679684  2.611368  2.544793  2.479916  2.416693  2.355081
##  [85]  2.295041  2.236531  2.179512  2.123948  2.069800  2.017032  1.965610
##  [92]  1.915498  1.866664  1.819075  1.772700  1.727506  1.683465  1.640547
##  [99]  1.598722  1.557964  1.518245
## 
## 
## [[22]]
## [[22]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[22]]$y
##   [1]  0.000000  3.960080  7.005229  9.326954 11.077107 12.376088 13.319352
##   [8] 13.982561 14.425684 14.696240 14.831875 14.862402 14.811424 14.697619
##  [15] 14.535759 14.337517 14.112111 13.866812 13.607348 13.338223 13.062971
##  [22] 12.784359 12.504545 12.225205 11.947634 11.672826 11.401535 11.134329
##  [29] 10.871625 10.613725 10.360837 10.113098  9.870588  9.633343  9.401362
##  [36]  9.174622  8.953075  8.736660  8.525304  8.318923  8.117430  7.920731
##  [43]  7.728729  7.541328  7.358428  7.179929  7.005733  6.835742  6.669859
##  [50]  6.507988  6.350034  6.195906  6.045512  5.898763  5.755572  5.615853
##  [57]  5.479524  5.346502  5.216707  5.090063  4.966491  4.845919  4.728273
##  [64]  4.613483  4.501479  4.392194  4.285562  4.181519  4.080001  3.980948
##  [71]  3.884299  3.789997  3.697984  3.608205  3.520606  3.435133  3.351736
##  [78]  3.270363  3.190965  3.113495  3.037906  2.964152  2.892189  2.821973
##  [85]  2.753461  2.686613  2.621388  2.557746  2.495650  2.435061  2.375942
##  [92]  2.318260  2.261977  2.207061  2.153478  2.101196  2.050184  2.000410
##  [99]  1.951844  1.904457  1.858221
## 
## 
## [[23]]
## [[23]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[23]]$y
##   [1]  0.000000  3.931058  6.979743  9.326147 11.113975 12.457882 13.449342
##   [8] 14.161322 14.652026 14.967867 15.145853 15.215484 15.200267 15.118931
##  [15] 14.986385 14.814496 14.612701 14.388498 14.147839 13.895443 13.635043
##  [22] 13.369591 13.101409 12.832324 12.563764 12.296841 12.032415 11.771145
##  [29] 11.513532 11.259948 11.010666 10.765880 10.525720 10.290266 10.059559
##  [36]  9.833610  9.612405  9.395912  9.184084  8.976864  8.774185  8.575975
##  [43]  8.382156  8.192650  8.007374  7.826245  7.649178  7.476089  7.306895
##  [50]  7.141512  6.979859  6.821853  6.667416  6.516467  6.368930  6.224729
##  [57]  6.083789  5.946038  5.811403  5.679814  5.551204  5.425505  5.302651
##  [64]  5.182579  5.065224  4.950527  4.838426  4.728863  4.621782  4.517124
##  [71]  4.414837  4.314866  4.217158  4.121663  4.028330  3.937110  3.847957
##  [78]  3.760822  3.675660  3.592426  3.511077  3.431571  3.353865  3.277918
##  [85]  3.203691  3.131145  3.060242  2.990944  2.923215  2.857021  2.792325
##  [92]  2.729094  2.667295  2.606895  2.547863  2.490168  2.433779  2.378668
##  [99]  2.324804  2.272160  2.220708
## 
## 
## [[24]]
## [[24]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[24]]$y
##   [1]  0.000000  4.200332  7.406936  9.834118 11.650400 12.988297 13.951993
##   [8] 14.623395 15.066884 15.333057 15.461668 15.483944 15.424406 15.302298
##  [15] 15.132720 14.927505 14.695925 14.445233 14.181096 13.907938 13.629199
##  [22] 13.347553 13.065067 12.783334 12.503575 12.226717 11.953459 11.684319
##  [29] 11.419677 11.159800 10.904872 10.655008 10.410272 10.170688  9.936248
##  [36]  9.706922  9.482660  9.263400  9.049068  8.839585  8.634863  8.434814
##  [43]  8.239346  8.048364  7.861776  7.679488  7.501405  7.327436  7.157489
##  [50]  6.991473  6.829300  6.670882  6.516134  6.364972  6.217314  6.073079
##  [57]  5.932188  5.794564  5.660132  5.528818  5.400549  5.275256  5.152869
##  [64]  5.033321  4.916547  4.802481  4.691062  4.582227  4.475917  4.372074
##  [71]  4.270640  4.171559  4.074777  3.980240  3.887896  3.797695  3.709587
##  [78]  3.623522  3.539455  3.457337  3.377125  3.298774  3.222241  3.147483
##  [85]  3.074460  3.003131  2.933457  2.865399  2.798920  2.733984  2.670554
##  [92]  2.608596  2.548075  2.488958  2.431213  2.374808  2.319711  2.265892
##  [99]  2.213322  2.161972  2.111813
## 
## 
## [[25]]
## [[25]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[25]]$y
##   [1]  0.000000  4.139149  7.321760  9.749047 11.580311 12.941657 13.932905
##   [8] 14.633058 15.104637 15.397110 15.549605 15.593071 15.551972 15.445643
##  [15] 15.289355 15.095163 14.872575 14.629085 14.370588 14.101721 13.826118
##  [22] 13.546625 13.265461 12.984352 12.704633 12.427330 12.153224 11.882907
##  [29] 11.616815 11.355269 11.098492 10.846637 10.599795 10.358017 10.121314
##  [36]  9.889672  9.663056  9.441415  9.224685  9.012794  8.805661  8.603203
##  [43]  8.405332  8.211958  8.022990  7.838337  7.657907  7.481609  7.309353
##  [50]  7.141050  6.976612  6.815952  6.658985  6.505628  6.355798  6.209416
##  [57]  6.066403  5.926681  5.790176  5.656813  5.526522  5.399230  5.274870
##  [64]  5.153373  5.034675  4.918710  4.805416  4.694731  4.586596  4.480951
##  [71]  4.377740  4.276906  4.178394  4.082151  3.988125  3.896265  3.806520
##  [78]  3.718843  3.633185  3.549500  3.467743  3.387869  3.309834  3.233597
##  [85]  3.159116  3.086351  3.015262  2.945810  2.877957  2.811668  2.746905
##  [92]  2.683635  2.621821  2.561432  2.502433  2.444793  2.388481  2.333466
##  [99]  2.279718  2.227208  2.175908
## 
## 
## [[26]]
## [[26]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[26]]$y
##   [1]  0.000000  3.820685  6.783485  9.061878 10.794690 12.093004 13.045697
##   [8] 13.723865 14.184375 14.472701 14.625195 14.670913 14.633064 14.530184
##  [15] 14.377065 14.185501 13.964894 13.722724 13.464939 13.196259 12.920422
##  [22] 12.640381 12.358463 12.076494 11.795897 11.517777 11.242986 10.972168
##  [29] 10.705810 10.444267 10.187793  9.936561  9.690679  9.450205  9.215158
##  [36]  8.985523  8.761264  8.542324  8.328633  8.120110  7.916666  7.718204
##  [43]  7.524627  7.335832  7.151715  6.972172  6.797100  6.626393  6.459950
##  [50]  6.297668  6.139447  5.985190  5.834798  5.688177  5.545234  5.405879
##  [57]  5.270021  5.137575  5.008454  4.882577  4.759861  4.640229  4.523602
##  [64]  4.409906  4.299066  4.191012  4.085674  3.982982  3.882872  3.785277
##  [71]  3.690136  3.597385  3.506966  3.418819  3.332888  3.249116  3.167450
##  [78]  3.087837  3.010224  2.934563  2.860803  2.788897  2.718799  2.650462
##  [85]  2.583843  2.518898  2.455586  2.393865  2.333696  2.275038  2.217856
##  [92]  2.162110  2.107766  2.054787  2.003140  1.952792  1.903708  1.855859
##  [99]  1.809212  1.763738  1.719406
## 
## 
## [[27]]
## [[27]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[27]]$y
##   [1]  0.000000  4.483508  7.856366 10.369533 12.217789 13.552297 14.490359
##   [8] 15.122989 15.520791 15.738525 15.818655 15.794092 15.690341 15.527148
##  [15] 15.319800 15.080113 14.817216 14.538151 14.248340 13.951950 13.652172
##  [22] 13.351444 13.051617 12.754089 12.459907 12.169845 11.884469 11.604182
##  [29] 11.329262 11.059889 10.796173 10.538163 10.285868 10.039263  9.798298
##  [36]  9.562906  9.333005  9.108504  8.889305  8.675304  8.466395  8.262471
##  [43]  8.063423  7.869142  7.679520  7.494451  7.313829  7.137550  6.965512
##  [50]  6.797614  6.633759  6.473850  6.317792  6.165494  6.016866  5.871820
##  [57]  5.730268  5.592129  5.457319  5.325758  5.197369  5.072074  4.949800
##  [64]  4.830473  4.714023  4.600380  4.489476  4.381246  4.275625  4.172551
##  [71]  4.071961  3.973796  3.877997  3.784509  3.693273  3.604238  3.517348
##  [78]  3.432554  3.349803  3.269048  3.190239  3.113330  3.038276  2.965030
##  [85]  2.893551  2.823794  2.755720  2.689286  2.624454  2.561185  2.499441
##  [92]  2.439185  2.380383  2.322998  2.266996  2.212344  2.159010  2.106961
##  [99]  2.056168  2.006599  1.958225
## 
## 
## [[28]]
## [[28]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[28]]$y
##   [1]  0.000000  4.380895  7.710121 10.217295 12.082398 13.446490 14.420117
##   [8] 15.089900 15.523703 15.774682 15.884462 15.885632 15.803691 15.658585
##  [15] 15.465905 15.237827 14.983854 14.711391 14.426202 14.132762 13.834539
##  [22] 13.534212 13.233840 12.935001 12.638892 12.346417 12.058248 11.774875
##  [29] 11.496650 11.223816 10.956528 10.694878 10.438906 10.188614  9.943972
##  [36]  9.704930  9.471420  9.243358  9.020656  8.803216  8.590936  8.383711
##  [43]  8.181434  7.983999  7.791297  7.603223  7.419669  7.240532  7.065708
##  [50]  6.895096  6.728597  6.566113  6.407548  6.252809  6.101804  5.954443
##  [57]  5.810640  5.670308  5.533365  5.399728  5.269318  5.142057  5.017869
##  [64]  4.896680  4.778418  4.663011  4.550392  4.440493  4.333248  4.228593
##  [71]  4.126465  4.026804  3.929550  3.834645  3.742031  3.651655  3.563461
##  [78]  3.477398  3.393412  3.311456  3.231478  3.153433  3.077272  3.002950
##  [85]  2.930424  2.859649  2.790584  2.723186  2.657417  2.593236  2.530605
##  [92]  2.469486  2.409844  2.351642  2.294846  2.239421  2.185335  2.132556
##  [99]  2.081051  2.030790  1.981743
## 
## 
## [[29]]
## [[29]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[29]]$y
##   [1]  0.000000  4.279735  7.490643  9.875463 11.622352 12.877146 13.752867
##   [8] 14.337092 14.697670 14.887151 14.946228 14.906396 14.792023 14.621950
##  [15] 14.410735 14.169614 13.907249 13.630309 13.343917 13.051999 12.757554
##  [22] 12.462863 12.169653 11.879218 11.592525 11.310279 11.032991 10.761018
##  [29] 10.494600 10.233888  9.978962  9.729851  9.486544  9.249000  9.017154
##  [36]  8.790927  8.570225  8.354949  8.144989  7.940236  7.740576  7.545895
##  [43]  7.356077  7.171009  6.990578  6.814671  6.643179  6.475993  6.313007
##  [50]  6.154119  5.999224  5.848226  5.701025  5.557527  5.417640  5.281272
##  [57]  5.148336  5.018746  4.892417  4.769267  4.649217  4.532188  4.418106
##  [64]  4.306894  4.198482  4.092799  3.989776  3.889346  3.791444  3.696006
##  [71]  3.602971  3.512277  3.423867  3.337682  3.253666  3.171765  3.091926
##  [78]  3.014096  2.938226  2.864265  2.792166  2.721882  2.653368  2.586577
##  [85]  2.521468  2.457998  2.396126  2.335811  2.277014  2.219697  2.163823
##  [92]  2.109356  2.056260  2.004500  1.954043  1.904856  1.856907  1.810165
##  [99]  1.764600  1.720182  1.676881
## 
## 
## [[30]]
## [[30]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[30]]$y
##   [1]  0.000000  4.599794  8.075641 10.676374 12.596328 13.987262 14.967672
##   [8] 15.630067 16.046653 16.273777 16.355398 16.325795 16.211687 16.033889
##  [15] 15.808604 15.548432 15.263161 14.960384 14.645978 14.324482 13.999394
##  [22] 13.673393 13.348527 13.026346 12.708017 12.394405 12.086144 11.783685
##  [29] 11.487340 11.197310 10.913716 10.636611 10.366000 10.101851  9.844103
##  [36]  9.592674  9.347464  9.108365  8.875259  8.648023  8.426530  8.210651
##  [43]  8.000256  7.795217  7.595405  7.400693  7.210956  7.026069  6.845913
##  [50]  6.670367  6.499317  6.332648  6.170249  6.012012  5.857831  5.707602
##  [57]  5.561224  5.418599  5.279631  5.144226  5.012293  4.883744  4.758491
##  [64]  4.636451  4.517540  4.401678  4.288788  4.178793  4.071619  3.967194
##  [71]  3.865447  3.766310  3.669715  3.575597  3.483893  3.394541  3.307481
##  [78]  3.222653  3.140002  3.059469  2.981003  2.904549  2.830055  2.757472
##  [85]  2.686751  2.617843  2.550703  2.485285  2.421544  2.359439  2.298926
##  [92]  2.239965  2.182516  2.126541  2.072001  2.018860  1.967082  1.916632
##  [99]  1.867476  1.819580  1.772913
## 
## 
## [[31]]
## [[31]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[31]]$y
##   [1]  0.000000  4.088705  7.256917  9.692820 11.546521 12.937744 13.961973
##   [8] 14.695348 15.198578 15.520060 15.698367 15.764234 15.742144 15.651592
##  [15] 15.508095 15.323993 15.109096 14.871195 14.616470 14.349817 14.075107
##  [22] 13.795397 13.513092 13.230079 12.947833 12.667499 12.389962 12.115898
##  [29] 11.845817 11.580100 11.319023 11.062779 10.811495 10.565249 10.324077
##  [36] 10.087984  9.856950  9.630936  9.409888  9.193741  8.982420  8.775847
##  [43]  8.573937  8.376603  8.183754  7.995301  7.811152  7.631216  7.455402
##  [50]  7.283620  7.115782  6.951801  6.791589  6.635062  6.482136  6.332731
##  [57]  6.186765  6.044161  5.904842  5.768732  5.635758  5.505848  5.378931
##  [64]  5.254940  5.133805  5.015463  4.899849  4.786899  4.676552  4.568749
##  [71]  4.463431  4.360541  4.260022  4.161820  4.065882  3.972156  3.880590
##  [78]  3.791135  3.703742  3.618363  3.534953  3.453465  3.373856  3.296082
##  [85]  3.220101  3.145871  3.073352  3.002505  2.933292  2.865673  2.799614
##  [92]  2.735077  2.672028  2.610433  2.550257  2.491469  2.434035  2.377926
##  [99]  2.323110  2.269558  2.217240
## 
## 
## [[32]]
## [[32]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[32]]$y
##   [1]  0.000000  4.420916  7.795099 10.348647 12.259272 13.666644 14.680523
##   [8] 15.387157 15.854308 16.135204 16.271646 16.296457 16.235396 16.108673
##  [15] 15.932135 15.718199 15.476588 15.214907 14.939095 14.653784 14.362580
##  [22] 14.068280 13.773050 13.478556 13.186075 12.896579 12.610799 12.329279
##  [29] 12.052414 11.780487 11.513689 11.252144 10.995918 10.745039 10.499501
##  [36] 10.259271 10.024301  9.794526  9.569871  9.350254  9.135587  8.925776
##  [43]  8.720728  8.520347  8.324535  8.133196  7.946234  7.763553  7.585058
##  [50]  7.410657  7.240257  7.073769  6.911105  6.752177  6.596900  6.445192
##  [57]  6.296971  6.152157  6.010672  5.872440  5.737387  5.605438  5.476524
##  [64]  5.350574  5.227521  5.107297  4.989838  4.875081  4.762962  4.653422
##  [71]  4.546401  4.441841  4.339686  4.239880  4.142370  4.047102  3.954025
##  [78]  3.863089  3.774244  3.687443  3.602638  3.519783  3.438833  3.359746
##  [85]  3.282477  3.206985  3.133230  3.061170  2.990768  2.921986  2.854785
##  [92]  2.789129  2.724984  2.662313  2.601084  2.541264  2.482819  2.425718
##  [99]  2.369930  2.315426  2.262175
## 
## 
## [[33]]
## [[33]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[33]]$y
##   [1]  0.000000  3.747521  6.628728  8.825323 10.481294 11.710736 12.604046
##   [8] 13.232838 13.653834 13.911957 14.042771 14.074428 14.029202 13.924710
##  [15] 13.774875 13.590698 13.380860 13.152204 12.910121 12.658848 12.401707
##  [22] 12.141300 11.879657 11.618352 11.358603 11.101346 10.847292 10.596973
##  [29] 10.350785 10.109014  9.871857  9.639446  9.411859  9.189131  8.971267
##  [36]  8.758246  8.550027  8.346555  8.147766  7.953583  7.763927  7.578714
##  [43]  7.397856  7.221263  7.048846  6.880514  6.716177  6.555746  6.399130
##  [50]  6.246244  6.097001  5.951315  5.809105  5.670288  5.534784  5.402515
##  [57]  5.273405  5.147378  5.024362  4.904284  4.787075  4.672667  4.560992
##  [64]  4.451986  4.345584  4.241726  4.140349  4.041395  3.944805  3.850524
##  [71]  3.758497  3.668669  3.580987  3.495401  3.411861  3.330317  3.250722
##  [78]  3.173029  3.097194  3.023170  2.950916  2.880389  2.811547  2.744351
##  [85]  2.678760  2.614738  2.552245  2.491246  2.431705  2.373587  2.316858
##  [92]  2.261485  2.207435  2.154677  2.103180  2.052913  2.003848  1.955956
##  [99]  1.909209  1.863578  1.819038
## 
## 
## [[34]]
## [[34]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[34]]$y
##   [1]  0.000000  4.301756  7.542345  9.960064 11.740216 13.026884 13.932086
##   [8] 14.542903 14.927014 15.137011 15.213747 15.188950 15.087248 14.927748
##  [15] 14.725261 14.491262 14.234630 13.962225 13.679338 13.390042 13.097461
##  [22] 12.803984 12.511430 12.221171 11.934238 11.651395 11.373198 11.100045
##  [29] 10.832210 10.569872 10.313135 10.062051  9.816625  9.576832  9.342622
##  [36]  9.113926  8.890662  8.672738  8.460055  8.252510  8.049995  7.852403
##  [43]  7.659624  7.471549  7.288070  7.109079  6.934470  6.764140  6.597985
##  [50]  6.435906  6.277803  6.123580  5.973143  5.826399  5.683258  5.543633
##  [57]  5.407437  5.274586  5.144998  5.018594  4.895295  4.775025  4.657709
##  [64]  4.543275  4.431653  4.322773  4.216568  4.112972  4.011922  3.913354
##  [71]  3.817208  3.723424  3.631944  3.542711  3.455671  3.370770  3.287954
##  [78]  3.207173  3.128376  3.051516  2.976544  2.903414  2.832080  2.762500
##  [85]  2.694628  2.628425  2.563847  2.500857  2.439414  2.379480  2.321019
##  [92]  2.263995  2.208371  2.154114  2.101190  2.049566  1.999211  1.950093
##  [99]  1.902181  1.855447  1.809861
## 
## 
## [[35]]
## [[35]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[35]]$y
##   [1]  0.000000  4.188685  7.400522  9.840931 11.672651 13.024589 13.998844
##   [8] 14.676270 15.120883 15.383356 15.503790 15.513907 15.438790 15.298263
##  [15] 15.107981 14.880300 14.624958 14.349623 14.060323 13.761786 13.457713
##  [22] 13.150991 12.843863 12.538062 12.234922 11.935457 11.640429 11.350406
##  [29] 11.065796 10.786887 10.513872 10.246867  9.985929  9.731071  9.482270
##  [36]  9.239475  9.002616  8.771607  8.546348  8.326735  8.112654  7.903989
##  [43]  7.700622  7.502432  7.309299  7.121103  6.937726  6.759048  6.584956
##  [50]  6.415333  6.250070  6.089054  5.932181  5.779343  5.630439  5.485368
##  [57]  5.344032  5.206336  5.072185  4.941491  4.814162  4.690114  4.569262
##  [64]  4.451523  4.336817  4.225067  4.116196  4.010130  3.906797  3.806127
##  [71]  3.708051  3.612502  3.519415  3.428726  3.340375  3.254300  3.170442
##  [78]  3.088746  3.009155  2.931615  2.856073  2.782477  2.710778  2.640926
##  [85]  2.572875  2.506577  2.441987  2.379061  2.317757  2.258033  2.199848
##  [92]  2.143162  2.087937  2.034135  1.981719  1.930654  1.880904  1.832437
##  [99]  1.785218  1.739217  1.694400
## 
## 
## [[36]]
## [[36]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[36]]$y
##   [1]  0.000000  4.133718  7.305282  9.716829 11.528524 12.867263 13.833571
##   [8] 14.507071 14.950817 15.214733 15.338334 15.352887 15.283122 15.148587
##  [15] 14.964727 14.743731 14.495214 14.226748 13.944287 13.652505 13.355061
##  [22] 13.054815 12.753986 12.454295 12.157065 11.863303 11.573771 11.289035
##  [29] 11.009505 10.735473 10.467133 10.204607  9.947957  9.697201  9.452321
##  [36]  9.213272  8.979990  8.752393  8.530389  8.313877  8.102751  7.896898
##  [43]  7.696206  7.500559  7.309842  7.123941  6.942739  6.766125  6.593987
##  [50]  6.426215  6.262700  6.103337  5.948023  5.796656  5.649136  5.505368
##  [57]  5.365255  5.228707  5.095632  4.965942  4.839552  4.716378  4.596339
##  [64]  4.479354  4.365346  4.254239  4.145961  4.040437  3.937600  3.837379
##  [71]  3.739710  3.644526  3.551765  3.461365  3.373265  3.287408  3.203736
##  [78]  3.122194  3.042727  2.965283  2.889810  2.816258  2.744578  2.674722
##  [85]  2.606644  2.540299  2.475643  2.412632  2.351225  2.291381  2.233060
##  [92]  2.176224  2.120834  2.066854  2.014248  1.962981  1.913019  1.864328
##  [99]  1.816876  1.770633  1.725566
## 
## 
## [[37]]
## [[37]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[37]]$y
##   [1]  0.000000  4.214764  7.444840  9.899656 11.744530 13.109952 14.098908
##   [8] 14.792670 15.255365 15.537583 15.679224 15.711750 15.659961 15.543395
##  [15] 15.377440 15.174204 14.943211 14.691938 14.426254 14.150756 13.869036
##  [22] 13.583895 13.297512 13.011572 12.727372 12.445905 12.167924 11.893990
##  [29] 11.624519 11.359808 11.100064 10.845421 10.595958 10.351712 10.112683
##  [36]  9.878849  9.650165  9.426572  9.207999  8.994369  8.785594  8.581587
##  [43]  8.382255  8.187504  7.997238  7.811364  7.629786  7.452409  7.279141
##  [50]  7.109889  6.944564  6.783075  6.625336  6.471260  6.320764  6.173765
##  [57]  6.030182  5.889937  5.752952  5.619151  5.488462  5.360812  5.236130
##  [64]  5.114347  4.995397  4.879213  4.765730  4.654887  4.546622  4.440875
##  [71]  4.337588  4.236702  4.138163  4.041916  3.947907  3.856085  3.766398
##  [78]  3.678798  3.593235  3.509661  3.428032  3.348301  3.270425  3.194360
##  [85]  3.120064  3.047496  2.976616  2.907384  2.839763  2.773714  2.709202
##  [92]  2.646190  2.584644  2.524529  2.465812  2.408461  2.352444  2.297730
##  [99]  2.244288  2.192089  2.141104
## 
## 
## [[38]]
## [[38]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[38]]$y
##   [1]  0.000000  4.287488  7.563111 10.043616 11.899804 13.266255 14.249000
##   [8] 14.931565 15.379738 15.645326 15.769113 15.783199 15.712837 15.577885
##  [15] 15.393948 15.173281 14.925499 14.658135 14.377087 14.086960 13.791341
##  [22] 13.493019 13.194154 12.896408 12.601054 12.309060 12.021152 11.737868
##  [29] 11.459597 11.186615 10.919104 10.657178 10.400894 10.150269  9.905285
##  [36]  9.665901  9.432055  9.203672  8.980666  8.762941  8.550400  8.342939
##  [43]  8.140453  7.942834  7.749975  7.561771  7.378114  7.198899  7.024024
##  [50]  6.853385  6.686883  6.524419  6.365897  6.211222  6.060302  5.913046
##  [57]  5.769366  5.629176  5.492391  5.358929  5.228708  5.101652  4.977683
##  [64]  4.856725  4.738707  4.623556  4.511203  4.401580  4.294621  4.190261
##  [71]  4.088436  3.989087  3.892151  3.797571  3.705289  3.615249  3.527398
##  [78]  3.441681  3.358047  3.276446  3.196827  3.119143  3.043347  2.969393
##  [85]  2.897236  2.826832  2.758139  2.691116  2.625721  2.561915  2.499660
##  [92]  2.438917  2.379651  2.321825  2.265404  2.210354  2.156642  2.104235
##  [99]  2.053101  2.003210  1.954532
## 
## 
## [[39]]
## [[39]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[39]]$y
##   [1]  0.000000  4.208809  7.421414  9.851068 11.665887 12.998387 13.953010
##   [8] 14.612061 15.040388 15.289073 15.398341 15.399861 15.318551 15.174007
##  [15] 14.981633 14.753524 14.499168 14.226000 13.939835 13.645211 13.345665
##  [22] 13.043938 12.742152 12.441936 12.144536 11.850892 11.561708 11.277501
##  [29] 10.998643 10.725388 10.457905 10.196291  9.940589  9.690801  9.446897
##  [36]  9.208821  8.976501  8.749850  8.528769  8.313154  8.102896  7.897881
##  [43]  7.697995  7.503121  7.313144  7.127948  6.947420  6.771445  6.599914
##  [50]  6.432717  6.269747  6.110898  5.956069  5.805158  5.658067  5.514701
##  [57]  5.374965  5.238769  5.106022  4.976637  4.850531  4.727619  4.607822
##  [64]  4.491059  4.377256  4.266335  4.158225  4.052855  3.950155  3.850057
##  [71]  3.752495  3.657406  3.564726  3.474394  3.386352  3.300540  3.216904
##  [78]  3.135386  3.055934  2.978496  2.903019  2.829456  2.757756  2.687874
##  [85]  2.619762  2.553376  2.488672  2.425609  2.364143  2.304234  2.245844
##  [92]  2.188934  2.133465  2.079402  2.026709  1.975352  1.925296  1.876508
##  [99]  1.828956  1.782610  1.737438
## 
## 
## [[40]]
## [[40]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[40]]$y
##   [1]  0.000000  4.365530  7.666795 10.140568 11.971429 13.303259 14.248202
##   [8] 14.893656 15.307725 15.543463 15.642193 15.636092 15.550203 15.404010
##  [15] 15.212659 14.987920 14.738927 14.472759 14.194898 13.909578 13.620060
##  [22] 13.328851 13.037866 12.748565 12.462049 12.179142 11.900453 11.626423
##  [29] 11.357365 11.093491 10.834934 10.581770 10.334028 10.091700  9.854754
##  [36]  9.623138  9.396783  9.175609  8.959531  8.748454  8.542283  8.340918
##  [43]  8.144258  7.952203  7.764653  7.581506  7.402664  7.228028  7.057504
##  [50]  6.890995  6.728409  6.569655  6.414643  6.263286  6.115498  5.971196
##  [57]  5.830297  5.692723  5.558393  5.427233  5.299168  5.174124  5.052030
##  [64]  4.932817  4.816417  4.702764  4.591792  4.483439  4.377642  4.274343
##  [71]  4.173480  4.074998  3.978839  3.884950  3.793276  3.703766  3.616367
##  [78]  3.531031  3.447709  3.366352  3.286916  3.209354  3.133622  3.059677
##  [85]  2.987478  2.916981  2.848149  2.780941  2.715318  2.651244  2.588682
##  [92]  2.527597  2.467952  2.409716  2.352853  2.297332  2.243122  2.190190
##  [99]  2.138508  2.088045  2.038773
## 
## 
## [[41]]
## [[41]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[41]]$y
##   [1]  0.000000  4.156162  7.353536  9.792940 11.633561 13.001581 13.997004
##   [8] 14.699081 15.170608 15.461329 15.610643 15.649749 15.603337 15.490945
##  [15] 15.328021 15.126772 14.896839 14.645825 14.379720 14.103235 13.820068
##  [22] 13.533117 13.244641 12.956398 12.669749 12.385738 12.105161 11.828617
##  [29] 11.556552 11.289286 11.027047 10.769984 10.518187 10.271703 10.030539
##  [36]  9.794675  9.564071  9.338669  9.118400  8.903182  8.692932  8.487556
##  [43]  8.286961  8.091051  7.899728  7.712893  7.530449  7.352298  7.178344
##  [50]  7.008491  6.842647  6.680718  6.522614  6.368246  6.217527  6.070372
##  [57]  5.926697  5.786420  5.649461  5.515743  5.385189  5.257724  5.133275
##  [64]  5.011772  4.893144  4.777323  4.664244  4.553841  4.446052  4.340813
##  [71]  4.238065  4.137750  4.039809  3.944186  3.850826  3.759676  3.670684
##  [78]  3.583798  3.498969  3.416147  3.335286  3.256339  3.179261  3.104007
##  [85]  3.030535  2.958801  2.888766  2.820388  2.753629  2.688449  2.624813
##  [92]  2.562683  2.502024  2.442800  2.384979  2.328526  2.273409  2.219597
##  [99]  2.167058  2.115764  2.065683
## 
## 
## [[42]]
## [[42]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[42]]$y
##   [1]  0.000000  3.998266  7.070294  9.410887 11.174301 12.482672 13.432697
##   [8] 14.100921 14.547929 14.821664 14.960057 14.993107 14.944534 14.833083
##  [15] 14.673561 14.477655 14.254582 14.011604 13.754436 13.487567 13.214514
##  [22] 12.938030 12.660261 12.382870 12.107143 11.834067 11.564391 11.298677
##  [29] 11.037340 10.780681 10.528908 10.282158 10.040512  9.804008  9.572649
##  [36]  9.346413  9.125257  8.909123  8.697940  8.491631  8.290110  8.093289
##  [43]  7.901074  7.713373  7.530089  7.351128  7.176394  7.005792  6.839230
##  [50]  6.676616  6.517857  6.362866  6.211553  6.063834  5.919624  5.778840
##  [57]  5.641402  5.507230  5.376248  5.248380  5.123553  5.001693  4.882731
##  [64]  4.766598  4.653227  4.542552  4.434509  4.329035  4.226070  4.125554
##  [71]  4.027429  3.931637  3.838124  3.746835  3.657717  3.570719  3.485790
##  [78]  3.402881  3.321944  3.242932  3.165799  3.090501  3.016994  2.945235
##  [85]  2.875183  2.806797  2.740038  2.674867  2.611245  2.549137  2.488506
##  [92]  2.429317  2.371536  2.315130  2.260065  2.206309  2.153832  2.102604
##  [99]  2.052594  2.003773  1.956114
## 
## 
## [[43]]
## [[43]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[43]]$y
##   [1]  0.000000  4.617405  8.134563 10.791672 12.776934 14.237793 15.289761
##   [8] 16.023317 16.509328 16.803289 16.948650 16.979422 16.922216 16.797851
##  [15] 16.622601 16.409182 16.167525 15.905372 15.628758 15.342376 15.049870
##  [22] 14.754060 14.457128 14.160746 13.866198 13.574456 13.286254 13.002135
##  [29] 12.722499 12.447629 12.177719 11.912897 11.653233 11.398759 11.149474
##  [36] 10.905352 10.666348 10.432406 10.203455  9.979421  9.760220  9.545767
##  [43]  9.335973  9.130750  8.930005  8.733649  8.541590  8.353740  8.170008
##  [50]  7.990308  7.814554  7.642659  7.474540  7.310117  7.149307  6.992033
##  [57]  6.838217  6.687783  6.540658  6.396768  6.256043  6.118414  5.983812
##  [64]  5.852170  5.723425  5.597512  5.474368  5.353934  5.236149  5.120955
##  [71]  5.008296  4.898114  4.790357  4.684971  4.581902  4.481102  4.382518
##  [78]  4.286104  4.191811  4.099592  4.009402  3.921196  3.834931  3.750563
##  [85]  3.668052  3.587355  3.508434  3.431249  3.355763  3.281937  3.209735
##  [92]  3.139121  3.070062  3.002521  2.936466  2.871865  2.808684  2.746894
##  [99]  2.686463  2.627361  2.569560
## 
## 
## [[44]]
## [[44]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[44]]$y
##   [1]  0.000000  4.254396  7.522624 10.012560 11.888696 13.281183 14.292995
##   [8] 15.005592 15.483402 15.777381 15.927812 15.966538 15.918720 15.804230
##  [15] 15.638756 15.434673 15.201737 14.947629 14.678392 14.398770 14.112481
##  [22] 13.822433 13.530891 13.239614 12.949960 12.662972 12.379442 12.099966
##  [29] 11.824986 11.554820 11.289691 11.029748 10.775079 10.525728 10.281703
##  [36] 10.042983  9.809527  9.581277  9.358164  9.140108  8.927024  8.718821
##  [43]  8.515405  8.316680  8.122551  7.932919  7.747687  7.566759  7.390039
##  [50]  7.217434  7.048849  6.884193  6.723378  6.566313  6.412914  6.263095
##  [57]  6.116773  5.973868  5.834300  5.697992  5.564867  5.434851  5.307873
##  [64]  5.183860  5.062745  4.944459  4.828936  4.716113  4.605925  4.498311
##  [71]  4.393212  4.290568  4.190323  4.092419  3.996803  3.903421  3.812220
##  [78]  3.723151  3.636162  3.551206  3.468234  3.387202  3.308062  3.230772
##  [85]  3.155287  3.081566  3.009568  2.939252  2.870578  2.803509  2.738007
##  [92]  2.674036  2.611559  2.550542  2.490950  2.432751  2.375912  2.320400
##  [99]  2.266186  2.213238  2.161527
## 
## 
## [[45]]
## [[45]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[45]]$y
##   [1]  0.000000  3.649633  6.497818  8.703639 10.394959 11.674540 12.624961
##   [8] 13.312575 13.790691 14.102126 14.281269 14.355727 14.347658 14.274840
##  [15] 14.151529 13.989151 13.796856 13.581966 13.350334 13.106632 12.854584
##  [22] 12.597152 12.336687 12.075047 11.813697 11.553785 11.296208 11.041657
##  [29] 10.790660 10.543618 10.300823 10.062489  9.828762  9.599732  9.375453
##  [36]  9.155943  8.941192  8.731174  8.525844  8.325145  8.129013  7.937374
##  [43]  7.750150  7.567261  7.388621  7.214145  7.043746  6.877338  6.714834
##  [50]  6.556147  6.401192  6.249886  6.102144  5.957886  5.817030  5.679499
##  [57]  5.545214  5.414100  5.286084  5.161092  5.039053  4.919898  4.803560
##  [64]  4.689972  4.579068  4.470787  4.365066  4.261844  4.161063  4.062664
##  [71]  3.966593  3.872792  3.781210  3.691794  3.604492  3.519254  3.436032
##  [78]  3.354778  3.275445  3.197989  3.122364  3.048527  2.976436  2.906050
##  [85]  2.837329  2.770233  2.704723  2.640762  2.578314  2.517343  2.457814
##  [92]  2.399692  2.342945  2.287539  2.233444  2.180628  2.129062  2.078714
##  [99]  2.029557  1.981563  1.934703
## 
## 
## [[46]]
## [[46]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[46]]$y
##   [1]  0.000000  4.370260  7.731532 10.296815 12.234537 13.677876 14.732126
##   [8] 15.480528 15.988883 16.309194 16.482555 16.541432 16.511464 16.412895
##  [15] 16.261700 16.070479 15.849163 15.605574 15.345864 15.074868 14.796376
##  [22] 14.513357 14.228127 13.942489 13.657839 13.375254 13.095555 12.819368
##  [29] 12.547157 12.279266 12.015941 11.757350 11.503603 11.254761 11.010851
##  [36] 10.771870 10.537794 10.308582 10.084179  9.864521  9.649537  9.439151
##  [43]  9.233283  9.031851  8.834770  8.641955  8.453321  8.268783  8.088257
##  [50]  7.911659  7.738906  7.569917  7.404611  7.242910  7.084735  6.930012
##  [57]  6.778666  6.630622  6.485810  6.344160  6.205602  6.070069  5.937496
##  [64]  5.807818  5.680972  5.556896  5.435529  5.316813  5.200690  5.087103
##  [71]  4.975996  4.867317  4.761010  4.657026  4.555312  4.455820  4.358501
##  [78]  4.263308  4.170193  4.079113  3.990021  3.902876  3.817633  3.734253
##  [85]  3.652694  3.572915  3.494880  3.418549  3.343884  3.270851  3.199413
##  [92]  3.129535  3.061183  2.994324  2.928925  2.864955  2.802382  2.741175
##  [99]  2.681305  2.622743  2.565460
## 
## 
## [[47]]
## [[47]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[47]]$y
##   [1]  0.000000  4.617307  8.049271 10.575514 12.410210 13.717392 14.622677
##   [8] 15.222254 15.589765 15.781581 15.840844 15.800556 15.685957 15.516339
##  [15] 15.306439 15.067506 14.808112 14.534789 14.252499 13.965003 13.675147
##  [22] 13.385073 13.096387 12.810281 12.527637 12.249094 11.975110 11.706004
##  [29] 11.441987 11.183191 10.929686 10.681498 10.438615 10.201003  9.968606
##  [36]  9.741356  9.519172  9.301969  9.089655  8.882137  8.679317  8.481098
##  [43]  8.287383  8.098075  7.913078  7.732297  7.555638  7.383009  7.214320
##  [50]  7.049481  6.888406  6.731010  6.577208  6.426919  6.280064  6.136563
##  [57]  5.996341  5.859323  5.725435  5.594606  5.466767  5.341848  5.219784
##  [64]  5.100509  4.983960  4.870074  4.758790  4.650049  4.543792  4.439964
##  [71]  4.338508  4.239371  4.142499  4.047840  3.955344  3.864963  3.776646
##  [78]  3.690347  3.606021  3.523621  3.443104  3.364427  3.287548  3.212425
##  [85]  3.139020  3.067291  2.997202  2.928714  2.861791  2.796397  2.732498
##  [92]  2.670059  2.609046  2.549428  2.491172  2.434247  2.378623  2.324270
##  [99]  2.271159  2.219262  2.168551
## 
## 
## [[48]]
## [[48]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[48]]$y
##   [1]  0.000000  3.834033  6.822217  9.134235 10.906070 12.246688 13.243402
##   [8] 13.966164 14.471013 14.802834 14.997570 15.084001 15.085159 15.019477
##  [15] 14.901695 14.743598 14.554599 14.342213 14.112435 13.870039 13.618823
##  [22] 13.361803 13.101368 12.839406 12.577404 12.316527 12.057683 11.801576
##  [29] 11.548744 11.299595 11.054433 10.813478 10.576883 10.344752 10.117143
##  [36]  9.894083  9.675574  9.461596  9.252115  9.047082  8.846441  8.650128
##  [43]  8.458074  8.270206  8.086449  7.906724  7.730953  7.559058  7.390959
##  [50]  7.226577  7.065835  6.908654  6.754960  6.604676  6.457729  6.314046
##  [57]  6.173555  6.036187  5.901872  5.770544  5.642137  5.516585  5.393826
##  [64]  5.273798  5.156440  5.041693  4.929499  4.819801  4.712544  4.607674
##  [71]  4.505137  4.404882  4.306858  4.211015  4.117305  4.025680  3.936094
##  [78]  3.848501  3.762858  3.679121  3.597247  3.517195  3.438925  3.362396
##  [85]  3.287570  3.214410  3.142877  3.072937  3.004553  2.937690  2.872316
##  [92]  2.808396  2.745899  2.684793  2.625046  2.566629  2.509512  2.453667
##  [99]  2.399063  2.345675  2.293476
## 
## 
## [[49]]
## [[49]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[49]]$y
##   [1]  0.000000  4.353418  7.687785 10.219943 12.121052 13.526205 14.542021
##   [8] 15.252631 15.724415 16.009735 16.149887 16.177429 16.118023 15.991885
##  [15] 15.814931 15.599684 15.355991 15.091584 14.812525 14.523562 14.228405
##  [22] 13.929944 13.630424 13.331579 13.034746 12.740943 12.450940 12.165312
##  [29] 11.884481 11.608747 11.338318 11.073325 10.813845 10.559908 10.311509
##  [36] 10.068619  9.831185  9.599141  9.372410  9.150904  8.934530  8.723191
##  [43]  8.516788  8.315217  8.118377  7.926165  7.738480  7.555218  7.376282
##  [50]  7.201570  7.030988  6.864438  6.701828  6.543064  6.388058  6.236721
##  [57]  6.088967  5.944712  5.803873  5.666369  5.532122  5.401055  5.273093
##  [64]  5.148162  5.026190  4.907108  4.790847  4.677340  4.566523  4.458330
##  [71]  4.352702  4.249575  4.148892  4.050595  3.954626  3.860931  3.769455
##  [78]  3.680147  3.592955  3.507829  3.424720  3.343579  3.264361  3.187020
##  [85]  3.111512  3.037792  2.965819  2.895551  2.826948  2.759971  2.694580
##  [92]  2.630738  2.568409  2.507557  2.448147  2.390144  2.333515  2.278228
##  [99]  2.224251  2.171553  2.120103
## 
## 
## [[50]]
## [[50]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[50]]$y
##   [1]  0.000000  3.825729  6.788901  9.066843 10.800766 12.103116 13.063424
##   [8] 13.752970 14.228497 14.535166 14.708918 14.778345 14.766191 14.690537
##  [15] 14.565754 14.403259 14.212113 13.999505 13.771129 13.531495 13.284164
##  [22] 13.031945 12.777049 12.521211 12.265784 12.011824 11.760144 11.511368
##  [29] 11.265970 11.024304 10.786627 10.553124 10.323919 10.099089  9.878676
##  [36]  9.662691  9.451124  9.243946  9.041118  8.842586  8.648291  8.458167
##  [43]  8.272146  8.090154  7.912116  7.737957  7.567601  7.400970  7.237988
##  [50]  7.078580  6.922670  6.770183  6.621048  6.475191  6.332543  6.193033
##  [57]  6.056593  5.923157  5.792658  5.665033  5.540219  5.418153  5.298776
##  [64]  5.182029  5.067853  4.956193  4.846992  4.740197  4.635755  4.533614
##  [71]  4.433724  4.336034  4.240497  4.147064  4.055690  3.966330  3.878938
##  [78]  3.793472  3.709888  3.628147  3.548206  3.470027  3.393571  3.318799
##  [85]  3.245674  3.174161  3.104223  3.035826  2.968937  2.903521  2.839546
##  [92]  2.776981  2.715795  2.655956  2.597436  2.540206  2.484237  2.429500
##  [99]  2.375970  2.323619  2.272422
## 
## 
## [[51]]
## [[51]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[51]]$y
##   [1]  0.000000  3.611056  6.414530  8.574369 10.221581 11.460843 12.375790
##   [8] 13.033233 13.486539 13.778328 13.942631 14.006611 13.991948 13.915933
##  [15] 13.792356 13.632203 13.444226 13.235385 13.011214 12.776105 12.533539
##  [22] 12.286268 12.036463 11.785833 11.535715 11.287150 11.040945 10.797720
##  [29] 10.557944 10.321968 10.090049  9.862368  9.639047  9.420163  9.205753
##  [36]  8.995826  8.790370  8.589353  8.392730  8.200447  8.012439  7.828639
##  [43]  7.648972  7.473363  7.301733  7.134002  6.970090  6.809918  6.653404
##  [50]  6.500471  6.351039  6.205031  6.062372  5.922985  5.786797  5.653736
##  [57]  5.523731  5.396713  5.272613  5.151365  5.032903  4.917165  4.804087
##  [64]  4.693609  4.585671  4.480215  4.377184  4.276521  4.178174  4.082087
##  [71]  3.988211  3.896493  3.806884  3.719336  3.633802  3.550234  3.468588
##  [78]  3.388820  3.310886  3.234744  3.160354  3.087674  3.016666  2.947290
##  [85]  2.879510  2.813289  2.748591  2.685381  2.623624  2.563288  2.504339
##  [92]  2.446745  2.390477  2.335502  2.281792  2.229316  2.178048  2.127959
##  [99]  2.079021  2.031209  1.984497
## 
## 
## [[52]]
## [[52]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[52]]$y
##   [1]  0.000000  3.954318  6.977999  9.269847 10.986636 12.251987 13.163366
##   [8] 13.797609 14.215281 14.464107 14.581689 14.597644 14.535286 14.412966
##  [15] 14.245115 14.043073 13.815745 13.570115 13.311653 13.044635 12.772396
##  [22] 12.497530 12.222047 11.947498 11.675072 11.405672 11.139980 10.878500
##  [29] 10.621599 10.369537 10.122489  9.880563  9.643817  9.412269  9.185905
##  [36]  8.964691  8.748571  8.537477  8.331332  8.130050  7.933541  7.741710
##  [43]  7.554462  7.371699  7.193322  7.019234  6.849337  6.683535  6.521734
##  [50]  6.363839  6.209758  6.059401  5.912679  5.769506  5.629797  5.493468
##  [57]  5.360439  5.230629  5.103961  4.980360  4.859752  4.742063  4.627224
##  [64]  4.515166  4.405821  4.299124  4.195011  4.093419  3.994287  3.897556
##  [71]  3.803167  3.711064  3.621191  3.533495  3.447923  3.364423  3.282945
##  [78]  3.203440  3.125861  3.050160  2.976293  2.904215  2.833882  2.765252
##  [85]  2.698285  2.632939  2.569176  2.506957  2.446244  2.387002  2.329195
##  [92]  2.272788  2.217747  2.164038  2.111631  2.060492  2.010592  1.961901
##  [99]  1.914388  1.868027  1.822788
## 
## 
## [[53]]
## [[53]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[53]]$y
##   [1]  0.000000  3.670245  6.487632  8.631464 10.243764 11.437018 12.300309
##   [8] 12.904172 13.304451 13.545346 13.661837 13.681590 13.626487 13.513820
##  [15] 13.357248 13.167554 12.953239 12.721000 12.476104 12.222687 11.963989
##  [22] 11.702540 11.440310 11.178824 10.919260 10.662515 10.409269 10.160030
##  [29]  9.915168  9.674950  9.439554  9.209097  8.983641  8.763209  8.547794
##  [36]  8.337363  8.131866  7.931240  7.735408  7.544290  7.357796  7.175836
##  [43]  6.998315  6.825137  6.656206  6.491426  6.330701  6.173937  6.021040
##  [50]  5.871917  5.726478  5.584634  5.446297  5.311383  5.179807  5.051487
##  [57]  4.926344  4.804299  4.685276  4.569201  4.456000  4.345604  4.237941
##  [64]  4.132946  4.030552  3.930694  3.833310  3.738338  3.645719  3.555395
##  [71]  3.467309  3.381404  3.297628  3.215928  3.136252  3.058550  2.982772
##  [78]  2.908873  2.836804  2.766520  2.697978  2.631135  2.565947  2.502374
##  [85]  2.440376  2.379915  2.320951  2.263448  2.207370  2.152681  2.099347
##  [92]  2.047335  1.996611  1.947144  1.898902  1.851856  1.805975  1.761231
##  [99]  1.717596  1.675041  1.633541
## 
## 
## [[54]]
## [[54]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[54]]$y
##   [1]  0.000000  3.985552  7.079524  9.462672 11.279499 12.645515 13.653046
##   [8] 14.375880 14.872979 15.191451 15.368922 15.435443 15.414999 15.326735
##  [15] 15.185916 15.004714 14.792824 14.557959 14.306253 14.042576 13.770785
##  [22] 13.493933 13.214426 12.934157 12.654607 12.376930 12.102019 11.830555
##  [29] 11.563059 11.299915 11.041405 10.787725 10.539009 10.295335 10.056742
##  [36]  9.823236  9.594798  9.371390  9.152958  8.939436  8.730751  8.526822
##  [43]  8.327564  8.132889  7.942705  7.756921  7.575446  7.398185  7.225048
##  [50]  7.055944  6.890782  6.729474  6.571932  6.418071  6.267805  6.121052
##  [57]  5.977732  5.837764  5.701070  5.567576  5.437205  5.309886  5.185547
##  [64]  5.064119  4.945534  4.829725  4.716628  4.606178  4.498315  4.392977
##  [71]  4.290106  4.189644  4.091534  3.995721  3.902152  3.810774  3.721536
##  [78]  3.634388  3.549280  3.466165  3.384997  3.305729  3.228318  3.152719
##  [85]  3.078891  3.006792  2.936380  2.867618  2.800466  2.734887  2.670843
##  [92]  2.608299  2.547219  2.487570  2.429318  2.372429  2.316873  2.262618
##  [99]  2.209634  2.157890  2.107358
## 
## 
## [[55]]
## [[55]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[55]]$y
##   [1]  0.000000  4.638648  8.070839 10.584115 12.398057 13.680337 14.558971
##   [8] 15.131688 15.473083 15.640080 15.676106 15.614281 15.479851 15.292049
##  [15] 15.065517 14.811391 14.538129 14.252146 13.958297 13.660245 13.360746
##  [22] 13.061862 12.765129 12.471676 12.182330 11.897683 11.618150 11.344011
##  [29] 11.075444 10.812551 10.555376 10.303919 10.058147  9.818005  9.583418
##  [36]  9.354301  9.130558  8.912087  8.698783  8.490538  8.287243  8.088789
##  [43]  7.895067  7.705968  7.521387  7.341217  7.165357  6.993704  6.826159
##  [50]  6.662624  6.503005  6.347209  6.195143  6.046719  5.901851  5.760453
##  [57]  5.622442  5.487737  5.356259  5.227931  5.102677  4.980424  4.861100
##  [64]  4.744635  4.630960  4.520009  4.411715  4.306017  4.202850  4.102156
##  [71]  4.003874  3.907946  3.814317  3.722931  3.633735  3.546675  3.461702
##  [78]  3.378764  3.297813  3.218802  3.141684  3.066413  2.992946  2.921239
##  [85]  2.851250  2.782938  2.716262  2.651184  2.587666  2.525669  2.465157
##  [92]  2.406095  2.348448  2.292183  2.237265  2.183663  2.131345  2.080281
##  [99]  2.030440  1.981794  1.934313
## 
## 
## [[56]]
## [[56]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[56]]$y
##   [1]  0.000000  4.269769  7.528027  9.992379 11.834096 13.187970 14.160072
##   [8] 14.833860 15.274993 15.535110 15.654818 15.666040 15.593859 15.457974
##  [15] 15.273849 15.053614 14.806773 14.540764 14.261401 13.973221 13.679752
##  [22] 13.383735 13.087286 12.792035 12.499227 12.209806 11.924479 11.643771
##  [29] 11.368057 11.097604 10.832586 10.573112 10.319234 10.070965  9.828287
##  [36]  9.591154  9.359506  9.133265  8.912346  8.696655  8.486093  8.280557
##  [43]  8.079942  7.884144  7.693055  7.506570  7.324584  7.146992  6.973693
##  [50]  6.804586  6.639571  6.478551  6.321431  6.168117  6.018518  5.872545
##  [57]  5.730111  5.591130  5.455518  5.323195  5.194080  5.068097  4.945168
##  [64]  4.825222  4.708184  4.593984  4.482555  4.373828  4.267738  4.164222
##  [71]  4.063216  3.964660  3.868494  3.774662  3.683105  3.593768  3.506599
##  [78]  3.421544  3.338552  3.257573  3.178559  3.101461  3.026233  2.952829
##  [85]  2.881206  2.811321  2.743130  2.676593  2.611671  2.548323  2.486512
##  [92]  2.426200  2.367350  2.309929  2.253900  2.199230  2.145886  2.093836
##  [99]  2.043049  1.993493  1.945139
## 
## 
## [[57]]
## [[57]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[57]]$y
##   [1]  0.000000  3.933610  6.905088  9.129192 10.773183 11.967321 12.813038
##   [8] 13.389310 13.757619 13.965817 14.051139 14.042551 13.962577 13.828722
##  [15] 13.654584 13.450714 13.225297 12.984669 12.733729 12.476257 12.215165
##  [22] 11.952683 11.690518 11.429964 11.171999 10.917352 10.666561 10.420014
##  [29] 10.177985  9.940658  9.708150  9.480522  9.257797  9.039966  8.826997
##  [36]  8.618840  8.415432  8.216700  8.022563  7.832939  7.647737  7.466868
##  [43]  7.290242  7.117765  6.949347  6.784898  6.624326  6.467545  6.314466
##  [50]  6.165004  6.019075  5.876597  5.737488  5.601670  5.469065  5.339598
##  [57]  5.213194  5.089782  4.969291  4.851652  4.736797  4.624661  4.515180
##  [64]  4.408290  4.303930  4.202041  4.102564  4.005441  3.910618  3.818040
##  [71]  3.727653  3.639406  3.553248  3.469130  3.387003  3.306820  3.228536
##  [78]  3.152105  3.077483  3.004628  2.933497  2.864051  2.796248  2.730051
##  [85]  2.665420  2.602320  2.540714  2.480566  2.421842  2.364508  2.308532
##  [92]  2.253880  2.200523  2.148428  2.097567  2.047910  1.999429  1.952095
##  [99]  1.905882  1.860763  1.816711
## 
## 
## [[58]]
## [[58]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[58]]$y
##   [1]  0.000000  4.724072  8.267189 10.898050 12.824816 14.208729 15.174679
##   [8] 15.819381 16.217720 16.427663 16.494070 16.451645 16.327222 16.141537
##  [15] 15.910603 15.646775 15.359573 15.056320 14.742641 14.422846 14.100225
##  [22] 13.777280 13.455902 13.137513 12.823171 12.513651 12.209513 11.911149
##  [29] 11.618823 11.332701 11.052873 10.779372 10.512188 10.251276  9.996570
##  [36]  9.747982  9.505413  9.268755  9.037890  8.812701  8.593063  8.378855
##  [43]  8.169950  7.966227  7.767563  7.573837  7.384929  7.200724  7.021105
##  [50]  6.845962  6.675182  6.508660  6.346288  6.187966  6.033591  5.883066
##  [57]  5.736296  5.593186  5.453646  5.317587  5.184922  5.055567  4.929439
##  [64]  4.806457  4.686543  4.569621  4.455615  4.344454  4.236066  4.130383
##  [71]  4.027335  3.926859  3.828890  3.733364  3.640222  3.549404  3.460851
##  [78]  3.374508  3.290318  3.208230  3.128189  3.050145  2.974048  2.899850
##  [85]  2.827503  2.756960  2.688178  2.621112  2.555719  2.491957  2.429786
##  [92]  2.369167  2.310059  2.252427  2.196232  2.141439  2.088013  2.035920
##  [99]  1.985127  1.935600  1.887310
## 
## 
## [[59]]
## [[59]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[59]]$y
##   [1]  0.000000  4.576855  8.053654 10.670248 12.614741 14.034638 15.045597
##   [8] 15.738291 16.183793 16.437800 16.543949 16.536416 16.441955 16.281499
##  [15] 16.071418 15.824499 15.550724 15.257871 14.951995 14.637796 14.318916
##  [22] 13.998165 13.677703 13.359181 13.043852 12.732657 12.426293 12.125269
##  [29] 11.829944 11.540563 11.257282 10.980187 10.709310 10.444643 10.186147
##  [36]  9.933757  9.687394  9.446961  9.212355  8.983466  8.760177  8.542371
##  [43]  8.329927  8.122724  7.920643  7.723563  7.531368  7.343939  7.161162
##  [50]  6.982924  6.809115  6.639626  6.474352  6.313187  6.156032  6.002786
##  [57]  5.853354  5.707640  5.565552  5.427001  5.291898  5.160158  5.031697
##  [64]  4.906434  4.784289  4.665185  4.549045  4.435797  4.325368  4.217688
##  [71]  4.112688  4.010303  3.910466  3.813115  3.718187  3.625623  3.535363
##  [78]  3.447350  3.361528  3.277842  3.196240  3.116670  3.039080  2.963422
##  [85]  2.889647  2.817709  2.747562  2.679161  2.612463  2.547426  2.484008
##  [92]  2.422168  2.361868  2.303069  2.245734  2.189826  2.135310  2.082152
##  [99]  2.030316  1.979771  1.930485
## 
## 
## [[60]]
## [[60]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[60]]$y
##   [1]  0.000000  3.888326  6.858189  9.106492 10.788365 12.026012 12.915695
##   [8] 13.533224 13.938300 14.177921 14.289082 14.300891 14.236243 14.113137
##  [15] 13.945714 13.745078 13.519936 13.277113 13.021944 12.758599 12.490323
##  [22] 12.219640 11.948501 11.678409 11.410515 11.145691 10.884595 10.627711
##  [29] 10.375389 10.127877  9.885340  9.647878  9.415542  9.188347  8.966276
##  [36]  8.749290  8.537332  8.330333  8.128216  7.930893  7.738275  7.550268
##  [43]  7.366775  7.187699  7.012943  6.842410  6.676004  6.513628  6.355188
##  [50]  6.200593  6.049750  5.902571  5.758967  5.618853  5.482145  5.348761
##  [57]  5.218620  5.091644  4.967757  4.846883  4.728949  4.613884  4.501619
##  [64]  4.392085  4.285215  4.180946  4.079214  3.979957  3.883115  3.788630
##  [71]  3.696443  3.606499  3.518744  3.433125  3.349588  3.268084  3.188564
##  [78]  3.110978  3.035280  2.961424  2.889365  2.819060  2.750465  2.683539
##  [85]  2.618242  2.554534  2.492375  2.431730  2.372560  2.314829  2.258504
##  [92]  2.203548  2.149931  2.097617  2.046577  1.996779  1.948192  1.900788
##  [99]  1.854537  1.809411  1.765384
## 
## 
## [[61]]
## [[61]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[61]]$y
##   [1]  0.000000  4.333103  7.660156 10.194029 12.102979 13.519985 14.550141
##   [8] 15.276492 15.764651 16.066448 16.222819 16.266085 16.221761 16.109976
##  [15] 15.946603 15.744156 15.512488 15.259350 14.990835 14.711723 14.425757
##  [22] 14.135864 13.844322 13.552900 13.262963 12.975559 12.691487 12.411347
##  [29] 12.135583 11.864518 11.598380 11.337321 11.081433 10.830764 10.585325
##  [36] 10.345100 10.110051  9.880126  9.655258  9.435372  9.220386  9.010213
##  [43]  8.804764  8.603947  8.407668  8.215833  8.028349  7.845123  7.666062
##  [50]  7.491075  7.320072  7.152964  6.989665  6.830089  6.674152  6.521772
##  [57]  6.372869  6.227363  6.085178  5.946238  5.810470  5.677801  5.548160
##  [64]  5.421479  5.297690  5.176727  5.058526  4.943024  4.830158  4.719870
##  [71]  4.612100  4.506790  4.403886  4.303330  4.205071  4.109055  4.015232
##  [78]  3.923550  3.833963  3.746420  3.660877  3.577287  3.495605  3.415789
##  [85]  3.337795  3.261582  3.187109  3.114337  3.043226  2.973739  2.905838
##  [92]  2.839488  2.774653  2.711298  2.649390  2.588896  2.529783  2.472019
##  [99]  2.415575  2.360419  2.306523
## 
## 
## [[62]]
## [[62]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[62]]$y
##   [1]  0.000000  4.450467  7.829690 10.372798 12.263794 13.646659 14.634047
##   [8] 15.314091 15.755727 16.012869 16.127671 16.133082 16.054846 15.913069
##  [15] 15.723446 15.498218 15.246922 14.976984 14.694174 14.402968 14.106831
##  [22] 13.808437 13.509841 13.212615 12.917955 12.626759 12.339700 12.057267
##  [29] 11.779813 11.507581 11.240732 10.979359 10.723507 10.473181 10.228357
##  [36]  9.988990  9.755016  9.526359  9.302933  9.084647  8.871405  8.663106
##  [43]  8.459651  8.260936  8.066859  7.877319  7.692214  7.511445  7.334913
##  [50]  7.162521  6.994174  6.829779  6.669243  6.512478  6.359395  6.209908
##  [57]  6.063934  5.921389  5.782195  5.646272  5.513543  5.383935  5.257372
##  [64]  5.133784  5.013102  4.895256  4.780180  4.667810  4.558081  4.450931
##  [71]  4.346300  4.244129  4.144359  4.046935  3.951801  3.858903  3.768189
##  [78]  3.679608  3.593109  3.508643  3.426163  3.345622  3.266974  3.190175
##  [85]  3.115181  3.041950  2.970441  2.900613  2.832426  2.765842  2.700824
##  [92]  2.637334  2.575336  2.514796  2.455679  2.397951  2.341581  2.286536
##  [99]  2.232785  2.180297  2.129043
## 
## 
## [[63]]
## [[63]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[63]]$y
##   [1]  0.000000  4.207285  7.459791  9.953168 11.843438 13.255010 14.287071
##   [8] 15.018694 15.512901 15.819915 15.979748 16.024265 15.978837 15.863655
##  [15] 15.694773 15.484955 15.244333 14.980947 14.701165 14.410023 14.111498
##  [22] 13.808719 13.504144 13.199693 12.896860 12.596798 12.300393 12.008312
##  [29] 11.721057 11.438990 11.162371 10.891372 10.626099 10.366608 10.112912
##  [36]  9.864992  9.622805  9.386288  9.155366  8.929949  8.709943  8.495246
##  [43]  8.285753  8.081354  7.881941  7.687404  7.497633  7.312517  7.131949
##  [50]  6.955821  6.784029  6.616467  6.453036  6.293633  6.138162  5.986527
##  [57]  5.838635  5.694393  5.553711  5.416504  5.282685  5.152171  5.024880
##  [64]  4.900733  4.779653  4.661564  4.546392  4.434066  4.324514  4.217669
##  [71]  4.113463  4.011832  3.912712  3.816040  3.721758  3.629804  3.540122
##  [78]  3.452656  3.367351  3.284154  3.203012  3.123875  3.046694  2.971419
##  [85]  2.898004  2.826402  2.756570  2.688464  2.622039  2.557257  2.494074
##  [92]  2.432453  2.372354  2.313740  2.256574  2.200821  2.146445  2.093413
##  [99]  2.041691  1.991246  1.942049
## 
## 
## [[64]]
## [[64]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[64]]$y
##   [1]  0.000000  3.943135  6.981596  9.302463 11.054593 12.356418 13.302151
##   [8] 13.966729 14.409745 14.678581 14.810899 14.836623 14.779522 14.658464
##  [15] 14.488421 14.281259 14.046378 13.791214 13.521639 13.242286 12.956797
##  [22] 12.668032 12.378224 12.089114 11.802045 11.518052 11.237919 10.962236
##  [29] 10.691436 10.425832 10.165637  9.910990  9.661970  9.418608  9.180899
##  [36]  8.948811  8.722287  8.501258  8.285639  8.075336  7.870250  7.670276
##  [43]  7.475307  7.285232  7.099942  6.919326  6.743273  6.571676  6.404426
##  [50]  6.241416  6.082544  5.927705  5.776800  5.629731  5.486401  5.346716
##  [57]  5.210585  5.077916  4.948624  4.822622  4.699828  4.580158  4.463536
##  [64]  4.349882  4.239121  4.131180  4.025988  3.923473  3.823569  3.726209
##  [71]  3.631327  3.538862  3.448751  3.360934  3.275353  3.191952  3.110674
##  [78]  3.031466  2.954274  2.879049  2.805738  2.734295  2.664670  2.596818
##  [85]  2.530695  2.466255  2.403455  2.342255  2.282613  2.224490  2.167847
##  [92]  2.112646  2.058851  2.006426  1.955335  1.905546  1.857024  1.809738
##  [99]  1.763656  1.718747  1.674982
## 
## 
## [[65]]
## [[65]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[65]]$y
##   [1]  0.000000  3.944921  6.940936  9.196346 10.874148 12.101874 12.979302
##   [8] 13.584485 13.978472 14.209003 14.313404 14.320849 14.254131 14.131052
##  [15] 13.965506 13.768331 13.547971 13.311000 13.062529 12.806521 12.546045
##  [22] 12.283472 12.020622 11.758889 11.499335 11.242758 10.989754 10.740761
##  [29] 10.496094 10.255972 10.020537  9.789877  9.564034  9.343015  9.126802
##  [36]  8.915356  8.708623  8.506540  8.309033  8.116023  7.927427  7.743159
##  [43]  7.563132  7.387257  7.215446  7.047611  6.883664  6.723518  6.567088
##  [50]  6.414289  6.265040  6.119259  5.976867  5.837785  5.701937  5.569249
##  [57]  5.439647  5.313060  5.189418  5.068653  4.950698  4.835487  4.722957
##  [64]  4.613045  4.505691  4.400836  4.298420  4.198388  4.100683  4.005253
##  [71]  3.912043  3.821002  3.732080  3.645227  3.560395  3.477538  3.396609
##  [78]  3.317563  3.240357  3.164948  3.091293  3.019352  2.949086  2.880455
##  [85]  2.813421  2.747947  2.683997  2.621535  2.560527  2.500939  2.442737
##  [92]  2.385890  2.330365  2.276133  2.223163  2.171426  2.120892  2.071535
##  [99]  2.023326  1.976239  1.930248
## 
## 
## [[66]]
## [[66]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[66]]$y
##   [1]  0.000000  4.651310  8.205307 10.895779 12.907280 14.385479 15.445345
##   [8] 16.177595 16.653804 16.930427 17.051982 17.053558 16.962807 16.801500
##  [15] 16.586772 16.332096 16.048055 15.742948 15.423278 15.094126 14.759453
##  [22] 14.422335 14.085155 13.749746 13.417507 13.089500 12.766518 12.449144
##  [29] 12.137798 11.832768 11.534243 11.242334 10.957090 10.678513 10.406567
##  [36] 10.141192  9.882303  9.629801  9.383576  9.143508  8.909473  8.681342
##  [43]  8.458984  8.242267  8.031061  7.825233  7.624654  7.429195  7.238730
##  [50]  7.053135  6.872289  6.696071  6.524365  6.357058  6.194036  6.035192
##  [57]  5.880419  5.729614  5.582674  5.439501  5.299999  5.164074  5.031634
##  [64]  4.902591  4.776857  4.654347  4.534978  4.418671  4.305347  4.194929
##  [71]  4.087342  3.982515  3.880377  3.780857  3.683890  3.589410  3.497353
##  [78]  3.407657  3.320262  3.235108  3.152137  3.071295  2.992526  2.915777
##  [85]  2.840997  2.768134  2.697140  2.627967  2.560568  2.494897  2.430911
##  [92]  2.368566  2.307820  2.248631  2.190961  2.134770  2.080020  2.026674
##  [99]  1.974696  1.924051  1.874705
## 
## 
## [[67]]
## [[67]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[67]]$y
##   [1]  0.000000  4.217616  7.395273  9.764535 11.506002 12.760504 13.637850
##   [8] 14.223641 14.584595 14.772699 14.828445 14.783360 14.661979 14.483380
##  [15] 14.262386 14.010505 13.736655 13.447737 13.149082 12.844794 12.538022
##  [22] 12.231172 11.926072 11.624100 11.326282 11.033375 10.745923 10.464308
##  [29] 10.188786  9.919513  9.656573  9.399993  9.149753  8.905803  8.668068
##  [36]  8.436453  8.210850  7.991142  7.777206  7.568913  7.366134  7.168736
##  [43]  6.976589  6.789560  6.607522  6.430345  6.257904  6.090076  5.926740
##  [50]  5.767777  5.613073  5.462514  5.315990  5.173394  5.034621  4.899568
##  [57]  4.768138  4.640231  4.515756  4.394618  4.276730  4.162004  4.050355
##  [64]  3.941701  3.835962  3.733059  3.632916  3.535460  3.440618  3.348320
##  [71]  3.258498  3.171086  3.086018  3.003233  2.922668  2.844265  2.767965
##  [78]  2.693711  2.621450  2.551127  2.482690  2.416090  2.351276  2.288200
##  [85]  2.226817  2.167081  2.108946  2.052372  1.997315  1.943735  1.891592
##  [92]  1.840848  1.791466  1.743408  1.696639  1.651125  1.606832  1.563727
##  [99]  1.521779  1.480956  1.441227
## 
## 
## [[68]]
## [[68]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[68]]$y
##   [1]  0.000000  4.346213  7.640934 10.116076 11.952883 13.292987 14.247058
##   [8] 14.901558 15.324030 15.567223 15.672327 15.671491 15.589803 15.446827
##  [15] 15.257814 15.034641 14.786549 14.520722 14.242734 13.956901 13.666560
##  [22] 13.374278 13.082028 12.791313 12.503273 12.218765 11.938424 11.662713
##  [29] 11.391962 11.126396 10.866162 10.611344 10.361976 10.118058  9.879561
##  [36]  9.646436  9.418616  9.196023  8.978571  8.766167  8.558715  8.356115
##  [43]  8.158266  7.965067  7.776416  7.592212  7.412354  7.236744  7.065285
##  [50]  6.897879  6.734434  6.574857  6.419058  6.266947  6.118438  5.973447
##  [57]  5.831891  5.693688  5.558759  5.427027  5.298416  5.172852  5.050264
##  [64]  4.930581  4.813734  4.699656  4.588281  4.479546  4.373387  4.269744
##  [71]  4.168558  4.069769  3.973321  3.879159  3.787228  3.697476  3.609851
##  [78]  3.524303  3.440782  3.359240  3.279631  3.201908  3.126028  3.051945
##  [85]  2.979618  2.909006  2.840066  2.772761  2.707050  2.642897  2.580264
##  [92]  2.519115  2.459416  2.401131  2.344228  2.288673  2.234434  2.181481
##  [99]  2.129783  2.079311  2.030034
## 
## 
## [[69]]
## [[69]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[69]]$y
##   [1]  0.000000  4.433043  7.829485 10.409222 12.345987 13.777039 14.810799
##   [8] 15.532905 16.010987 16.298440 16.437414 16.461169 16.395943 16.262421
##  [15] 16.076904 15.852229 15.598494 15.323634 15.033876 14.734100 14.428119
##  [22] 14.118904 13.808766 13.499490 13.192447 12.888686 12.588998 12.293971
##  [29] 12.004037 11.719503 11.440577 11.167392 10.900021 10.638492 10.382795
##  [36] 10.132894  9.888731  9.650233  9.417315  9.189882  8.967834  8.751065
##  [43]  8.539469  8.332937  8.131357  7.934620  7.742617  7.555240  7.372380
##  [50]  7.193934  7.019796  6.849866  6.684042  6.522228  6.364327  6.210246
##  [57]  6.059893  5.913178  5.770013  5.630313  5.493995  5.360976  5.231178
##  [64]  5.104521  4.980931  4.860333  4.742654  4.627825  4.515775  4.406439
##  [71]  4.299749  4.195643  4.094057  3.994931  3.898205  3.803821  3.711722
##  [78]  3.621853  3.534160  3.448590  3.365092  3.283615  3.204112  3.126533
##  [85]  3.050833  2.976965  2.904886  2.834552  2.765921  2.698952  2.633605
##  [92]  2.569839  2.507618  2.446903  2.387658  2.329847  2.273436  2.218391
##  [99]  2.164679  2.112267  2.061124
## 
## 
## [[70]]
## [[70]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[70]]$y
##   [1]  0.000000  4.573295  7.989380 10.515091 12.356321 13.671920 14.584434
##   [8] 15.188383 15.556664 15.745477 15.798142 15.748038 15.620869 15.436421
##  [15] 15.209909 14.953025 14.674741 14.381931 14.079851 13.772511 13.462958
##  [22] 13.153496 12.845863 12.541352 12.240922 11.945269 11.654892 11.370135
##  [29] 11.091225 10.818300 10.551431 10.290635 10.035892  9.787151  9.544341
##  [36]  9.307374  9.076151  8.850565  8.630502  8.415847  8.206481  8.002285
##  [43]  7.803141  7.608930  7.419535  7.234841  7.054733  6.879102  6.707836
##  [50]  6.540830  6.377978  6.219178  6.064329  5.913334  5.766098  5.622526
##  [57]  5.482529  5.346017  5.212903  5.083103  4.956536  4.833119  4.712775
##  [64]  4.595428  4.481003  4.369426  4.260628  4.154539  4.051092  3.950220
##  [71]  3.851860  3.755949  3.662426  3.571232  3.482309  3.395599  3.311049
##  [78]  3.228604  3.148212  3.069822  2.993384  2.918849  2.846170  2.775300
##  [85]  2.706196  2.638812  2.573105  2.509035  2.446560  2.385641  2.326239
##  [92]  2.268316  2.211835  2.156760  2.103057  2.050691  1.999629  1.949839
##  [99]  1.901288  1.853946  1.807783
## 
## 
## [[71]]
## [[71]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[71]]$y
##   [1]  0.000000  4.348758  7.645991 10.124678 11.966611 13.313628 14.276386
##   [8] 14.941209 15.375426 15.631544 15.750504 15.764216 15.697548 15.569868
##  [15] 15.396255 15.188440 14.955543 14.704645 14.441239 14.169576 13.892942
##  [22] 13.613868 13.334298 13.055716 12.779253 12.505758 12.235867 11.970047
##  [29] 11.708633 11.451861 11.199887 10.952808 10.710673 10.473496 10.241262
##  [36] 10.013936  9.791469  9.573797  9.360849  9.152549  8.948815  8.749561
##  [43]  8.554701  8.364147  8.177813  7.995609  7.817449  7.643246  7.472916
##  [50]  7.306374  7.143538  6.984327  6.828660  6.676460  6.527650  6.382155
##  [57]  6.239902  6.100818  5.964834  5.831880  5.701890  5.574796  5.450535
##  [64]  5.329043  5.210259  5.094123  4.980575  4.869558  4.761016  4.654893
##  [71]  4.551136  4.449691  4.350507  4.253534  4.158723  4.066025  3.975393
##  [78]  3.886782  3.800145  3.715440  3.632623  3.551651  3.472485  3.395083
##  [85]  3.319407  3.245417  3.173077  3.102349  3.033197  2.965587  2.899484
##  [92]  2.834854  2.771665  2.709885  2.649482  2.590425  2.532684  2.476230
##  [99]  2.421035  2.367070  2.314308
## 
## 
## [[72]]
## [[72]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[72]]$y
##   [1]  0.000000  4.354460  7.684927 10.211976 12.109070 13.512609 14.529846
##   [8] 15.245105 15.724672 16.020652 16.173989 16.216851 16.174506 16.066792
##  [15] 15.909278 15.714175 15.491054 15.247410 14.989106 14.720721 14.445825
##  [22] 14.167196 13.886989 13.606869 13.328117 13.051712 12.778399 12.508732
##  [29] 12.243125 11.981875 11.725192 11.473214 11.226029 10.983678 10.746174
##  [36] 10.513503 10.285630 10.062509  9.844081  9.630278  9.421027  9.216251
##  [43]  9.015870  8.819801  8.627961  8.440267  8.256633  8.076979  7.901219
##  [50]  7.729274  7.561063  7.396505  7.235524  7.078043  6.923986  6.773280
##  [57]  6.625852  6.481631  6.340549  6.202536  6.067526  5.935455  5.806258
##  [64]  5.679873  5.556239  5.435295  5.316984  5.201248  5.088032  4.977279
##  [71]  4.868937  4.762954  4.659277  4.557858  4.458646  4.361593  4.266653
##  [78]  4.173779  4.082927  3.994053  3.907113  3.822066  3.738870  3.657485
##  [85]  3.577871  3.499991  3.423805  3.349278  3.276373  3.205056  3.135290
##  [92]  3.067043  3.000282  2.934974  2.871088  2.808592  2.747456  2.687651
##  [99]  2.629149  2.571919  2.515935
## 
## 
## [[73]]
## [[73]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[73]]$y
##   [1]  0.000000  3.659369  6.491682  8.666567 10.319222 11.557398 12.466960
##   [8] 13.116326 13.560005 13.841414 13.995132 14.048688 14.023993 13.938479
##  [15] 13.806009 13.637600 13.442000 13.226151 12.995554 12.754562 12.506614
##  [22] 12.254422 12.000117 11.745368 11.491477 11.239455 10.990079 10.743943
##  [29] 10.501492 10.263057 10.028877  9.799117  9.573884  9.353242  9.137217
##  [36]  8.925808  8.718994  8.516735  8.318980  8.125665  7.936723  7.752078
##  [43]  7.571653  7.395367  7.223136  7.054878  6.890508  6.729943  6.573101
##  [50]  6.419898  6.270253  6.124086  5.981319  5.841874  5.705675  5.572647
##  [57]  5.442718  5.315815  5.191869  5.070811  4.952575  4.837095  4.724307
##  [64]  4.614148  4.506557  4.401474  4.298841  4.198601  4.100699  4.005079
##  [71]  3.911689  3.820476  3.731390  3.644381  3.559401  3.476403  3.395340
##  [78]  3.316167  3.238840  3.163316  3.089554  3.017511  2.947148  2.878426
##  [85]  2.811307  2.745753  2.681727  2.619194  2.558119  2.498469  2.440209
##  [92]  2.383308  2.327734  2.273455  2.220442  2.168666  2.118097  2.068707
##  [99]  2.020468  1.973355  1.927340
## 
## 
## [[74]]
## [[74]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[74]]$y
##   [1]  0.000000  3.886718  6.910574  9.245109 11.029340 12.374616 13.370117
##   [8] 14.087257 14.583220 14.903785 15.085590 15.157956 15.144335 15.063482
##  [15] 14.930386 14.757018 14.552938 14.325765 14.081574 13.825194 13.560465
##  [22] 13.290427 13.017486 12.743538 12.470072 12.198251 11.928977 11.662944
##  [29] 11.400681 11.142584 10.888944 10.639967 10.395795 10.156513  9.922169
##  [36]  9.692775  9.468319  9.248767  9.034070  8.824170  8.618996  8.418471
##  [43]  8.222515  8.031043  7.843967  7.661199  7.482649  7.308229  7.137849
##  [50]  6.971421  6.808856  6.650070  6.494975  6.343490  6.195530  6.051017
##  [57]  5.909870  5.772011  5.637366  5.505860  5.377419  5.251973  5.129453
##  [64]  5.009790  4.892917  4.778771  4.667287  4.558403  4.452059  4.348196
##  [71]  4.246756  4.147682  4.050920  3.956414  3.864114  3.773966  3.685922
##  [78]  3.599931  3.515947  3.433922  3.353810  3.275568  3.199151  3.124516
##  [85]  3.051623  2.980430  2.910898  2.842989  2.776663  2.711885  2.648618
##  [92]  2.586827  2.526478  2.467537  2.409970  2.353747  2.298835  2.245205
##  [99]  2.192825  2.141668  2.091704
## 
## 
## [[75]]
## [[75]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[75]]$y
##   [1]  0.000000  4.660482  8.257145 11.009950 13.093873 14.648089 15.783284
##   [8] 16.587480 17.130662 17.468465 17.645103 17.695707 17.648180 17.524672
##  [15] 17.342761 17.116385 16.856585 16.572102 16.269845 15.955267 15.632663
##  [22] 15.305409 14.976150 14.646948 14.319407 13.994763 13.673961 13.357717
##  [29] 13.046566 12.740896 12.440983 12.147013 11.859100 11.577305 11.301644
##  [36] 11.032099 10.768627 10.511164 10.259631 10.013939  9.773987  9.539672
##  [43]  9.310884  9.087511  8.869439  8.656555  8.448745  8.245894  8.047891
##  [50]  7.854625  7.665984  7.481863  7.302155  7.126756  6.955564  6.788480
##  [57]  6.625406  6.466246  6.310907  6.159298  6.011330  5.866916  5.725970
##  [64]  5.588409  5.454152  5.323120  5.195236  5.070424  4.948610  4.829723
##  [71]  4.713691  4.600447  4.489924  4.382055  4.276778  4.174031  4.073751
##  [78]  3.975881  3.880363  3.787138  3.696154  3.607355  3.520690  3.436107
##  [85]  3.353556  3.272988  3.194356  3.117613  3.042713  2.969614  2.898270
##  [92]  2.828640  2.760683  2.694359  2.629628  2.566452  2.504794  2.444617
##  [99]  2.385886  2.328567  2.272624
## 
## 
## [[76]]
## [[76]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[76]]$y
##   [1]  0.000000  4.336717  7.624852 10.096141 11.931559 13.272431 14.229107
##   [8] 14.887740 15.315586 15.565135 15.677348 15.684176 15.610536 15.475851
##  [15] 15.295251 15.080515 14.840805 14.583238 14.313338 14.035382 13.752674
##  [22] 13.467760 13.182596 12.898674 12.617128 12.338810 12.064355 11.794229
##  [29] 11.528765 11.268193 11.012665 10.762272 10.517056 10.277026 10.042158
##  [36]  9.812413  9.587733  9.368047  9.153277  8.943340  8.738147  8.537606
##  [43]  8.341623  8.150106  7.962959  7.780088  7.601401  7.426805  7.256210
##  [50]  7.089525  6.926664  6.767539  6.612066  6.460161  6.311745  6.166736
##  [57]  6.025057  5.886632  5.751387  5.619249  5.490145  5.364008  5.240768
##  [64]  5.120359  5.002717  4.887778  4.775479  4.665760  4.558562  4.453826
##  [71]  4.351498  4.251520  4.153839  4.058402  3.965158  3.874057  3.785048
##  [78]  3.698085  3.613119  3.530106  3.449000  3.369757  3.292335  3.216692
##  [85]  3.142787  3.070579  3.000031  2.931104  2.863760  2.797964  2.733679
##  [92]  2.670871  2.609506  2.549551  2.490974  2.433743  2.377826  2.323194
##  [99]  2.269818  2.217667  2.166715
## 
## 
## [[77]]
## [[77]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[77]]$y
##   [1]  0.000000  4.584570  8.058196 10.666421 12.601021 14.011767 15.015615
##   [8] 15.703882 16.147860 16.403193 16.513302 16.512059 16.425880 16.275353
##  [15] 16.076516 15.841852 15.581069 15.301708 15.009617 14.709322 14.404319
##  [22] 14.097299 13.790326 13.484972 13.182430 12.883594 12.589129 12.299517
##  [29] 12.015102 11.736121 11.462723 11.194997 10.932979 10.676669 10.426034
##  [36] 10.181024  9.941568  9.707586  9.478987  9.255674  9.037547  8.824501
##  [43]  8.616432  8.413232  8.214796  8.021019  7.831795  7.647022  7.466598
##  [50]  7.290422  7.118397  6.950426  6.786415  6.626271  6.469904  6.317224
##  [57]  6.168147  6.022586  5.880459  5.741686  5.606187  5.473885  5.344705
##  [64]  5.218573  5.095418  4.975169  4.857758  4.743118  4.631183  4.521889
##  [71]  4.415175  4.310979  4.209242  4.109906  4.012914  3.918211  3.825743
##  [78]  3.735457  3.647302  3.561227  3.477184  3.395124  3.315001  3.236768
##  [85]  3.160382  3.085798  3.012975  2.941870  2.872443  2.804655  2.738466
##  [92]  2.673840  2.610738  2.549126  2.488968  2.430229  2.372877  2.316878
##  [99]  2.262201  2.208814  2.156687
## 
## 
## [[78]]
## [[78]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[78]]$y
##   [1]  0.000000  4.064361  7.159958  9.495996 11.236989 12.512295 13.423614
##   [8] 14.050895 14.456973 14.691224 14.792434 14.791065 14.711025 14.571073
##  [15] 14.385919 14.167082 13.923581 13.662462 13.389227 13.108156 12.822577
##  [22] 12.535066 12.247607 11.961725 11.678578 11.399041 11.123766 10.853229
##  [29] 10.587772 10.327628 10.072949  9.823821  9.580281  9.342328  9.109930
##  [36]  8.883035  8.661571  8.445456  8.234598  8.028897  7.828251  7.632553
##  [43]  7.441695  7.255569  7.074066  6.897078  6.724498  6.556221  6.392143
##  [50]  6.232161  6.076176  5.924089  5.775804  5.631227  5.490265  5.352831
##  [57]  5.218834  5.088191  4.960817  4.836630  4.715552  4.597504  4.482411
##  [64]  4.370199  4.260796  4.154131  4.050136  3.948745  3.849892  3.753513
##  [71]  3.659547  3.567933  3.478613  3.391529  3.306625  3.223846  3.143140
##  [78]  3.064454  2.987738  2.912942  2.840019  2.768921  2.699604  2.632021
##  [85]  2.566131  2.501890  2.439257  2.378192  2.318656  2.260610  2.204018
##  [92]  2.148842  2.095047  2.042599  1.991465  1.941610  1.893003  1.845613
##  [99]  1.799410  1.754363  1.710444
## 
## 
## [[79]]
## [[79]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[79]]$y
##   [1]  0.000000  4.444685  7.820657 10.360981 12.248431 13.626328 14.607041
##   [8] 15.278662 15.710237 15.955872 16.057956 16.049684 15.957042 15.800361
##  [15] 15.595536 15.354988 15.088406 14.803344 14.505680 14.199977 13.889768
##  [22] 13.577780 13.266111 12.956361 12.649745 12.347175 12.049327 11.756693
##  [29] 11.469621 11.188348 10.913023 10.643729 10.380495 10.123314  9.872146
##  [36]  9.626927  9.387579  9.154008  8.926112  8.703783  8.486909  8.275372
##  [43]  8.069057  7.867845  7.671619  7.480261  7.293658  7.111694  6.934258
##  [50]  6.761239  6.592530  6.428025  6.267621  6.111215  5.958710  5.810008
##  [57]  5.665016  5.523640  5.385792  5.251383  5.120327  4.992542  4.867945
##  [64]  4.746458  4.628002  4.512503  4.399885  4.290079  4.183012  4.078617
##  [71]  3.976828  3.877579  3.780807  3.686450  3.594448  3.504742  3.417275
##  [78]  3.331990  3.248834  3.167753  3.088696  3.011612  2.936452  2.863167
##  [85]  2.791711  2.722039  2.654105  2.587867  2.523282  2.460309  2.398907
##  [92]  2.339038  2.280663  2.223745  2.168247  2.114135  2.061372  2.009927
##  [99]  1.959765  1.910856  1.863167
## 
## 
## [[80]]
## [[80]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[80]]$y
##   [1]  0.000000  3.959351  7.046401  9.433919 11.260886 12.639095 13.658457
##   [8] 14.391275 14.895680 15.218402 15.396996 15.461633 15.436544 15.341180
##  [15] 15.191147 14.998956 14.774631 14.526193 14.260056 13.981336 13.694110
##  [22] 13.401619 13.106431 12.810574 12.515642 12.222883 11.933263 11.647527
##  [29] 11.366238 11.089817 10.818572 10.552717 10.292394 10.037687  9.788637
##  [36]  9.545245  9.307485  9.075311  8.848657  8.627445  8.411588  8.200990
##  [43]  7.995551  7.795167  7.599731  7.409135  7.223272  7.042033  6.865311
##  [50]  6.692998  6.524991  6.361184  6.201477  6.045770  5.893963  5.745961
##  [57]  5.601671  5.460999  5.323857  5.190155  5.059809  4.932735  4.808851
##  [64]  4.688077  4.570335  4.455550  4.343646  4.234553  4.128199  4.024516
##  [71]  3.923437  3.824897  3.728831  3.635178  3.543877  3.454869  3.368096
##  [78]  3.283503  3.201034  3.120637  3.042258  2.965849  2.891358  2.818738
##  [85]  2.747943  2.678925  2.611641  2.546046  2.482100  2.419759  2.358984
##  [92]  2.299735  2.241975  2.185665  2.130770  2.077253  2.025080  1.974218
##  [99]  1.924634  1.876294  1.829169
## 
## 
## [[81]]
## [[81]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[81]]$y
##   [1]  0.000000  3.976273  7.017224  9.323144 11.051848 12.327676 13.248580
##   [8] 13.891721 14.317868 14.574873 14.700410 14.724127 14.669355 14.554436
##  [15] 14.393792 14.198747 13.978191 13.739091 13.486904 13.225895 12.959388
##  [22] 12.689972 12.419653 12.149979 11.882140 11.617042 11.355368 11.097627
##  [29] 10.844193 10.595330 10.351221 10.111981  9.877676  9.648332  9.423943
##  [36]  9.204482  8.989903  8.780145  8.575139  8.374808  8.179068  7.987833
##  [43]  7.801015  7.618523  7.440266  7.266152  7.096092  6.929996  6.767774
##  [50]  6.609339  6.454604  6.303486  6.155901  6.011768  5.871006  5.733537
##  [57]  5.599285  5.468175  5.340134  5.215090  5.092974  4.973716  4.857250
##  [64]  4.743511  4.632436  4.523961  4.418025  4.314571  4.213539  4.114872
##  [71]  4.018516  3.924416  3.832520  3.742775  3.655132  3.569541  3.485955
##  [78]  3.404326  3.324608  3.246757  3.170729  3.096481  3.023972  2.953161
##  [85]  2.884008  2.816474  2.750522  2.686114  2.623214  2.561787  2.501799
##  [92]  2.443215  2.386003  2.330131  2.275567  2.222281  2.170243  2.119423
##  [99]  2.069793  2.021326  1.973993
## 
## 
## [[82]]
## [[82]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[82]]$y
##   [1]  0.000000  4.305688  7.573190 10.030500 11.856024 13.189355 14.139700
##   [8] 14.792490 15.214542 15.458107 15.564031 15.564240 15.483674 15.341808
##  [15] 15.153841 14.931625 14.684393 14.419329 14.142014 13.856779 13.566970
##  [22] 13.275170 12.983363 12.693063 12.405420 12.121300 11.841345 11.566024
##  [29] 11.295672 11.030519 10.770713 10.516341 10.267439 10.024009  9.786021
##  [36]  9.553426  9.326158  9.104139  8.887282  8.675495  8.468678  8.266733
##  [43]  8.069557  7.877047  7.689102  7.505619  7.326497  7.151637  6.980939
##  [50]  6.814307  6.651645  6.492862  6.337865  6.186565  6.038874  5.894707
##  [57]  5.753981  5.616613  5.482523  5.351634  5.223869  5.099154  4.977416
##  [64]  4.858584  4.742589  4.629363  4.518841  4.410956  4.305648  4.202853
##  [71]  4.102513  4.004568  3.908961  3.815637  3.724541  3.635620  3.548822
##  [78]  3.464096  3.381393  3.300664  3.221862  3.144942  3.069859  2.996568
##  [85]  2.925026  2.855193  2.787027  2.720488  2.655538  2.592139  2.530253
##  [92]  2.469845  2.410879  2.353320  2.297136  2.242294  2.188760  2.136505
##  [99]  2.085497  2.035707  1.987106
## 
## 
## [[83]]
## [[83]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[83]]$y
##   [1]  0.000000  4.104515  7.209782  9.537913 11.262097 12.517366 13.408990
##   [8] 14.019030 14.411441 14.636061 14.731717 14.728647 14.650393 14.515274
##  [15] 14.337538 14.128256 13.896024 13.647509 13.387870 13.121098 12.850267
##  [22] 12.577740 12.305326 12.034402 11.766010 11.500929 11.239738 10.982854
##  [29] 10.730577 10.483107 10.240576 10.003057  9.770579  9.543139  9.320710
##  [36]  9.103243  8.890677  8.682940  8.479951  8.281626  8.087876  7.898608
##  [43]  7.713730  7.533150  7.356773  7.184507  7.016261  6.851943  6.691465
##  [50]  6.534739  6.381678  6.232198  6.086216  5.943651  5.804424  5.668456
##  [57]  5.535672  5.405998  5.279360  5.155689  5.034914  4.916968  4.801785
##  [64]  4.689300  4.579449  4.472172  4.367408  4.265098  4.165184  4.067611
##  [71]  3.972324  3.879269  3.788394  3.699647  3.612980  3.528343  3.445688
##  [78]  3.364970  3.286143  3.209162  3.133984  3.060568  2.988871  2.918854
##  [85]  2.850478  2.783703  2.718492  2.654809  2.592618  2.531883  2.472572
##  [92]  2.414650  2.358084  2.302844  2.248898  2.196215  2.144767  2.094524
##  [99]  2.045458  1.997541  1.950747
## 
## 
## [[84]]
## [[84]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[84]]$y
##   [1]  0.000000  4.248265  7.477390  9.912285 11.728602 13.063514 14.024138
##   [8] 14.694113 15.138737 15.408983 15.544635 15.576736 15.529502 15.421818
##  [15] 15.268403 15.080723 14.867707 14.636296 14.391885 14.138659 13.879860
##  [22] 13.617990 13.354980 13.092310 12.831109 12.572237 12.316338 12.063892
##  [29] 11.815249 11.570661 11.330301 11.094282 10.862669 10.635494 10.412759
##  [36] 10.194446  9.980521  9.770937  9.565638  9.364562  9.167643  8.974809
##  [43]  8.785988  8.601107  8.420089  8.242861  8.069346  7.899472  7.733165
##  [50]  7.570350  7.410958  7.254917  7.102158  6.952613  6.806214  6.662896
##  [57]  6.522595  6.385247  6.250790  6.119164  5.990309  5.864167  5.740681
##  [64]  5.619796  5.501455  5.385607  5.272197  5.161176  5.052493  4.946098
##  [71]  4.841944  4.739983  4.640169  4.542457  4.446802  4.353162  4.261493
##  [78]  4.171755  4.083907  3.997908  3.913720  3.831306  3.750626  3.671646
##  [85]  3.594329  3.518640  3.444544  3.372010  3.301002  3.231490  3.163441
##  [92]  3.096826  3.031613  2.967774  2.905279  2.844100  2.784209  2.725579
##  [99]  2.668184  2.611998  2.556994
## 
## 
## [[85]]
## [[85]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[85]]$y
##   [1]  0.000000  4.482611  7.863693 10.390767 12.256222 13.609582 14.567048
##   [8] 15.218912 15.635322 15.870770 15.967572 15.958583 15.869298 15.719495
##  [15] 15.524507 15.296211 15.043800 14.774379 14.493433 14.205185 13.912882
##  [22] 13.619009 13.325462 13.033681 12.744747 12.459469 12.178441 11.902093
##  [29] 11.630726 11.364546 11.103679 10.848195 10.598119 10.353443 10.114131
##  [36]  9.880130  9.651371  9.427775  9.209255  8.995720  8.787073  8.583217
##  [43]  8.384052  8.189479  7.999399  7.813713  7.632323  7.455134  7.282049
##  [50]  7.112976  6.947824  6.786503  6.628924  6.475002  6.324652  6.177791
##  [57]  6.034340  5.894218  5.757350  5.623660  5.493073  5.365519  5.240926
##  [64]  5.119226  5.000352  4.884238  4.770821  4.660037  4.551825  4.446126
##  [71]  4.342882  4.242035  4.143530  4.047312  3.953329  3.861528  3.771859
##  [78]  3.684272  3.598718  3.515152  3.433526  3.353795  3.275916  3.199845
##  [85]  3.125541  3.052962  2.982069  2.912822  2.845182  2.779114  2.714579
##  [92]  2.651544  2.589972  2.529829  2.471084  2.413702  2.357653  2.302905
##  [99]  2.249429  2.197195  2.146173
## 
## 
## [[86]]
## [[86]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[86]]$y
##   [1]  0.000000  4.017105  7.050092  9.319236 10.995950 12.213609 13.075986
##   [8] 13.663808 14.039868 14.252998 14.341164 14.333876 14.254061 14.119524
##  [15] 13.944086 13.738463 13.510960 13.268001 13.014553 12.754443 12.490618
##  [22] 12.225335 11.960318 11.696877 11.435999 11.178421 10.924686 10.675189
##  [29] 10.430204 10.189921  9.954456  9.723874  9.498199  9.277423  9.061517
##  [36]  8.850430  8.644102  8.442459  8.245424  8.052912  7.864836  7.681108
##  [43]  7.501636  7.326330  7.155099  6.987853  6.824504  6.664963  6.509144
##  [50]  6.356961  6.208332  6.063174  5.921408  5.782953  5.647735  5.515676
##  [57]  5.386705  5.260748  5.137736  5.017600  4.900272  4.785688  4.673783
##  [64]  4.564494  4.457761  4.353523  4.251723  4.152303  4.055208  3.960384
##  [71]  3.867776  3.777334  3.689007  3.602745  3.518500  3.436226  3.355875
##  [78]  3.277403  3.200766  3.125921  3.052826  2.981440  2.911724  2.843637
##  [85]  2.777143  2.712204  2.648783  2.586845  2.526356  2.467281  2.409587
##  [92]  2.353243  2.298216  2.244475  2.191992  2.140735  2.090677  2.041790
##  [99]  1.994046  1.947418  1.901881
## 
## 
## [[87]]
## [[87]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[87]]$y
##   [1]  0.000000  4.389232  7.753790 10.310797 12.231843 13.652524 14.679983
##   [8] 15.398868 15.876045 16.164323 16.305400 16.332188 16.270659 16.141296
##  [15] 15.960244 15.740221 15.491236 15.221158 14.936164 14.641095 14.339739
##  [22] 14.035047 13.729315 13.424320 13.121428 12.821683 12.525875 12.234593
##  [29] 11.948270 11.667214 11.391636 11.121673 10.857399 10.598846 10.346009
##  [36] 10.098853  9.857326  9.621358  9.390865  9.165758  8.945939  8.731306
##  [43]  8.521756  8.317182  8.117476  7.922533  7.732245  7.546506  7.365213
##  [50]  7.188262  7.015552  6.846984  6.682459  6.521883  6.365161  6.212202
##  [57]  6.062916  5.917216  5.775016  5.636232  5.500781  5.368586  5.239566
##  [64]  5.113647  4.990754  4.870813  4.753755  4.639510  4.528011  4.419191
##  [71]  4.312986  4.209333  4.108171  4.009441  3.913083  3.819041  3.727259
##  [78]  3.637683  3.550259  3.464937  3.381665  3.300394  3.221076  3.143665
##  [85]  3.068114  2.994379  2.922416  2.852182  2.783636  2.716738  2.651447
##  [92]  2.587725  2.525535  2.464840  2.405603  2.347789  2.291365  2.236298
##  [99]  2.182553  2.130100  2.078908
## 
## 
## [[88]]
## [[88]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[88]]$y
##   [1]  0.000000  4.096092  7.258327  9.680055 11.515003 12.885388 13.888364
##   [8] 14.601151 15.085120 15.389036 15.551637 15.603692 15.569626 15.468825
##  [15] 15.316668 15.125343 14.904507 14.661801 14.403267 14.133671 13.856770
##  [22] 13.575517 13.292225 13.008703 12.726356 12.446269 12.169276 11.896007
##  [29] 11.626938 11.362414 11.102684 10.847919 10.598227 10.353667 10.114261
##  [36]  9.880002  9.650860  9.426787  9.207721  8.993590  8.784315  8.579812
##  [43]  8.379991  8.184762  7.994032  7.807707  7.625693  7.447898  7.274229
##  [50]  7.104593  6.938901  6.777063  6.618991  6.464600  6.313806  6.166525
##  [57]  6.022676  5.882181  5.744961  5.610940  5.480045  5.352202  5.227341
##  [64]  5.105392  4.986288  4.869961  4.756349  4.645386  4.537012  4.431166
##  [71]  4.327789  4.226824  4.128215  4.031905  3.937843  3.845975  3.756250
##  [78]  3.668618  3.583031  3.499440  3.417800  3.338064  3.260189  3.184130
##  [85]  3.109845  3.037294  2.966435  2.897229  2.829638  2.763624  2.699150
##  [92]  2.636180  2.574679  2.514613  2.455948  2.398651  2.342692  2.288038
##  [99]  2.234659  2.182525  2.131608
## 
## 
## [[89]]
## [[89]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[89]]$y
##   [1]  0.000000  3.554054  6.291950  8.383456  9.963406 11.138885 11.994925
##   [8] 12.599042 13.004837 13.254850 13.382839 13.415584 13.374320 13.275882
##  [15] 13.133602 12.958039 12.757542 12.538709 12.306747 12.065759 11.818969
##  [22] 11.568907 11.317551 11.066440 10.816766 10.569448 10.325184 10.084502
##  [29]  9.847793  9.615341  9.387343  9.163932  8.945188  8.731147  8.521818
##  [36]  8.317182  8.117200  7.921821  7.730981  7.544609  7.362626  7.184951
##  [43]  7.011499  6.842182  6.676913  6.515603  6.358164  6.204509  6.054551
##  [50]  5.908204  5.765384  5.626009  5.489996  5.357266  5.227741  5.101345
##  [57]  4.978002  4.857639  4.740184  4.625569  4.513723  4.404581  4.298078
##  [64]  4.194149  4.092733  3.993769  3.897198  3.802961  3.711003  3.621269
##  [71]  3.533704  3.448257  3.364876  3.283510  3.204113  3.126635  3.051030
##  [78]  2.977254  2.905262  2.835010  2.766458  2.699562  2.634285  2.570586
##  [85]  2.508427  2.447771  2.388582  2.330824  2.274463  2.219465  2.165797
##  [92]  2.113426  2.062322  2.012453  1.963790  1.916304  1.869966  1.824749
##  [99]  1.780625  1.737568  1.695553
## 
## 
## [[90]]
## [[90]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[90]]$y
##   [1]  0.000000  4.966477  8.700425 11.481104 13.525071 15.000255 16.036854
##   [8] 16.735805 17.175347 17.416111 17.505072 17.478612 17.364898 17.185727
##  [15] 16.957952 16.694595 16.405707 16.099037 15.780544 15.454808 15.125332
##  [22] 14.794790 14.465214 14.138134 13.814697 13.495753 13.181921 12.873640
##  [29] 12.571215 12.274845 11.984649 11.700682 11.422955 11.151440 10.886084
##  [36] 10.626815 10.373543 10.126170  9.884590  9.648690  9.418356  9.193471
##  [43]  8.973917  8.759576  8.550332  8.346068  8.146669  7.952024  7.762021
##  [50]  7.576551  7.395508  7.218787  7.046286  6.877905  6.713546  6.553112
##  [57]  6.396512  6.243653  6.094446  5.948805  5.806643  5.667878  5.532430
##  [64]  5.400218  5.271165  5.145197  5.022238  4.902218  4.785066  4.670714
##  [71]  4.559095  4.450143  4.343794  4.239987  4.138661  4.039756  3.943215
##  [78]  3.848981  3.756999  3.667215  3.579577  3.494033  3.410534  3.329030
##  [85]  3.249473  3.171818  3.096019  3.022031  2.949811  2.879317  2.810508
##  [92]  2.743343  2.677783  2.613790  2.551327  2.490356  2.430842  2.372750
##  [99]  2.316047  2.260698  2.206673
## 
## 
## [[91]]
## [[91]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[91]]$y
##   [1]  0.000000  3.794933  6.732767  8.988295 10.701073 11.982518 12.921569
##   [8] 13.589212 14.042097 14.325422 14.475242 14.520306 14.483532 14.383180
##  [15] 14.233792 14.046940 13.831824 13.595752 13.344524 13.082730 12.814001
##  [22] 12.541198 12.266573 11.991887 11.718514 11.447517 11.179713 10.915722
##  [29] 10.656009 10.400914 10.150680  9.905473  9.665395  9.430502  9.200811
##  [36]  8.976312  8.756967  8.542724  8.333517  8.129270  7.929897  7.735309
##  [43]  7.545412  7.360110  7.179305  7.002899  6.830794  6.662891  6.499094
##  [50]  6.339306  6.183432  6.031380  5.883059  5.738378  5.597249  5.459586
##  [57]  5.325306  5.194326  5.066565  4.941944  4.820388  4.701820  4.586167
##  [64]  4.473359  4.363324  4.255996  4.151308  4.049194  3.949592  3.852440
##  [71]  3.757677  3.665245  3.575087  3.487146  3.401368  3.317701  3.236091
##  [78]  3.156489  3.078845  3.003111  2.929240  2.857185  2.786903  2.718350
##  [85]  2.651484  2.586262  2.522644  2.460592  2.400065  2.341028  2.283442
##  [92]  2.227274  2.172487  2.119047  2.066922  2.016079  1.966487  1.918115
##  [99]  1.870933  1.824911  1.780021
## 
## 
## [[92]]
## [[92]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[92]]$y
##   [1]  0.000000  3.873097  6.840895  9.095370 10.788213 12.039274 12.943235
##   [8] 13.574887 13.993299 14.245111 14.367145 14.388458 14.331975 14.215771
##  [15] 14.054093 13.858157 13.636792 13.396933 13.144025 12.882335 12.615197
##  [22] 12.345210 12.074394 11.804311 11.536163 11.270866 11.009112 10.751419
##  [29] 10.498163 10.249613 10.005953  9.767300  9.533718  9.305232  9.081833
##  [36]  8.863492  8.650158  8.441767  8.238245  8.039510  7.845475  7.656049
##  [43]  7.471137  7.290644  7.114475  6.942534  6.774725  6.610954  6.451128
##  [50]  6.295154  6.142942  5.994404  5.849451  5.708000  5.569965  5.435266
##  [57]  5.303822  5.175555  5.050388  4.928248  4.809061  4.692755  4.579262
##  [64]  4.468513  4.360442  4.254985  4.152078  4.051659  3.953669  3.858049
##  [71]  3.764741  3.673690  3.584841  3.498141  3.413538  3.330980  3.250420
##  [78]  3.171808  3.095097  3.020241  2.947196  2.875917  2.806362  2.738489
##  [85]  2.672258  2.607629  2.544563  2.483022  2.422969  2.364369  2.307186
##  [92]  2.251386  2.196936  2.143802  2.091954  2.041359  1.991989  1.943812
##  [99]  1.896800  1.850926  1.806160
## 
## 
## [[93]]
## [[93]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[93]]$y
##   [1]  0.000000  4.320288  7.643201 10.178413 12.091942 13.515232 14.552344
##   [8] 15.285651 15.780348 16.088027 16.249507 16.297073 16.256253 16.147227
##  [15] 15.985934 15.784959 15.554233 15.301580 15.033162 14.753822 14.467360
##  [22] 14.176750 13.884315 13.591861 13.300784 13.012160 12.726810 12.445351
##  [29] 12.168246 11.895830 11.628340 11.365937 11.108720 10.856743 10.610021
##  [36] 10.368540 10.132267  9.901149  9.675122  9.454111  9.238035  9.026807
##  [43]  8.820338  8.618535  8.421305  8.228552  8.040183  7.856105  7.676223
##  [50]  7.500446  7.328683  7.160845  6.996843  6.836592  6.680007  6.527005
##  [57]  6.377505  6.231427  6.088693  5.949227  5.812955  5.679803  5.549700
##  [64]  5.422578  5.298366  5.177000  5.058413  4.942542  4.829326  4.718703
##  [71]  4.610613  4.505000  4.401806  4.300975  4.202454  4.106190  4.012131
##  [78]  3.920227  3.830428  3.742685  3.656953  3.573184  3.491335  3.411360
##  [85]  3.333217  3.256864  3.182260  3.109365  3.038140  2.968546  2.900547
##  [92]  2.834105  2.769185  2.705752  2.643772  2.583212  2.524040  2.466222
##  [99]  2.409729  2.354531  2.300596
## 
## 
## [[94]]
## [[94]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[94]]$y
##   [1]  0.000000  4.301440  7.547308  9.974686 11.767864 13.070086 13.992679
##   [8] 14.622158 15.025743 15.255657 15.352458 15.347638 15.265641 15.125432
##  [15] 14.941713 14.725877 14.486745 14.231134 13.964309 13.690326 13.412304
##  [22] 13.132632 12.853135 12.575197 12.299862 12.027909 11.759914 11.496295
##  [29] 11.237344 10.983262 10.734176 10.490156 10.251231 10.017396  9.788621
##  [36]  9.564858  9.346045  9.132109  8.922971  8.718544  8.518740  8.323468
##  [43]  8.132636  7.946150  7.763919  7.585850  7.411851  7.241833  7.075707
##  [50]  6.913386  6.754783  6.599815  6.448400  6.300456  6.155904  6.014668
##  [57]  5.876671  5.741839  5.610100  5.481384  5.355620  5.232741  5.112681
##  [64]  4.995376  4.880762  4.768778  4.659362  4.552458  4.448006  4.345950
##  [71]  4.246237  4.148811  4.053620  3.960613  3.869740  3.780953  3.694202
##  [78]  3.609442  3.526626  3.445711  3.366652  3.289407  3.213935  3.140194
##  [85]  3.068145  2.997749  2.928968  2.861766  2.796105  2.731951  2.669269
##  [92]  2.608025  2.548186  2.489720  2.432596  2.376782  2.322249  2.268967
##  [99]  2.216907  2.166042  2.116344
## 
## 
## [[95]]
## [[95]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[95]]$y
##   [1]  0.000000  4.062628  7.172009  9.530578 11.298253 12.601354 13.539648
##   [8] 14.191923 14.620386 14.874143 14.991947 15.004371 14.935526 14.804416
##  [15] 14.626010 14.412092 14.171925 13.912787 13.640385 13.359185 13.072675
##  [22] 12.783570 12.493978 12.205523 11.919452 11.636713 11.358019 11.083900
##  [29] 10.814738 10.550802 10.292274 10.039264  9.791828  9.549980  9.313704
##  [36]  9.082954  8.857671  8.637776  8.423183  8.213797  8.009518  7.810243
##  [43]  7.615864  7.426274  7.241365  7.061031  6.885163  6.713657  6.546409
##  [50]  6.383315  6.224274  6.069189  5.917963  5.770499  5.626707  5.486495
##  [57]  5.349774  5.216459  5.086465  4.959708  4.836110  4.715591  4.598075
##  [64]  4.483488  4.371755  4.262807  4.156573  4.052987  3.951982  3.853495
##  [71]  3.757461  3.663821  3.572514  3.483483  3.396670  3.312021  3.229482
##  [78]  3.148999  3.070522  2.994001  2.919387  2.846632  2.775691  2.706517
##  [85]  2.639068  2.573299  2.509169  2.446638  2.385664  2.326211  2.268239
##  [92]  2.211711  2.156593  2.102848  2.050442  1.999343  1.949517  1.900932
##  [99]  1.853559  1.807366  1.762324
## 
## 
## [[96]]
## [[96]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[96]]$y
##   [1]  0.000000  3.618814  6.426029  8.586270 10.231141 11.465839 12.374450
##   [8] 13.024170 13.468693 13.750917 13.905103 13.958610 13.933277 13.846528
##  [15] 13.712259 13.541544 13.343202 13.124250 12.890264 12.645667 12.393964
##  [22] 12.137925 11.879732 11.621099 11.363365 11.107573 10.854525 10.604836
##  [29] 10.358968 10.117266  9.879976  9.647271  9.419263  9.196018  8.977563
##  [36]  8.763897  8.554997  8.350819  8.151310  7.956404  7.766028  7.580102
##  [43]  7.398543  7.221266  7.048184  6.879207  6.714247  6.553216  6.396025
##  [50]  6.242586  6.092815  5.946626  5.803935  5.664661  5.528724  5.396044
##  [57]  5.266544  5.140150  5.016786  4.896381  4.778865  4.664168  4.552223
##  [64]  4.442963  4.336326  4.232248  4.130667  4.031524  3.934760  3.840319
##  [71]  3.748144  3.658181  3.570378  3.484682  3.401043  3.319411  3.239738
##  [78]  3.161978  3.086084  3.012012  2.939718  2.869158  2.800293  2.733080
##  [85]  2.667481  2.603456  2.540967  2.479979  2.420454  2.362359  2.305657
##  [92]  2.250317  2.196304  2.143589  2.092138  2.041922  1.992912  1.945078
##  [99]  1.898392  1.852827  1.808355
## 
## 
## [[97]]
## [[97]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[97]]$y
##   [1]  0.000000  4.155729  7.295061  9.644689 11.381217 12.642214 13.534836
##   [8] 14.142534 14.530292 14.748700 14.837134 14.826232 14.739824 14.596434
##  [15] 14.410453 14.193054 13.952901 13.696705 13.429654 13.155754 12.878086
##  [22] 12.599015 12.320347 12.043454 11.769370 11.498867 11.232513 10.970719
##  [29] 10.713772 10.461867 10.215124  9.973608  9.737341  9.506310  9.280482
##  [36]  9.059799  8.844194  8.633587  8.427892  8.227018  8.030868  7.839346
##  [43]  7.652353  7.469792  7.291562  7.117567  6.947710  6.781895  6.620029
##  [50]  6.462020  6.307777  6.157212  6.010237  5.866769  5.726723  5.590019
##  [57]  5.456577  5.326319  5.199170  5.075056  4.953905  4.835645  4.720208
##  [64]  4.607526  4.497534  4.390168  4.285365  4.183064  4.083205  3.985729
##  [71]  3.890581  3.797704  3.707044  3.618548  3.532165  3.447844  3.365536
##  [78]  3.285193  3.206768  3.130215  3.055489  2.982548  2.911347  2.841847
##  [85]  2.774005  2.707783  2.643142  2.580044  2.518453  2.458331  2.399645
##  [92]  2.342360  2.286443  2.231860  2.178580  2.126573  2.075806  2.026252
##  [99]  1.977881  1.930664  1.884575
## 
## 
## [[98]]
## [[98]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[98]]$y
##   [1]  0.000000  3.976381  7.021663  9.334071 11.070047 12.353053 13.280512
##   [8] 13.929287 14.360018 14.620530 14.748535 14.773766 14.719648 14.604638
##  [15] 14.443262 14.246950 14.024687 13.783526 13.528999 13.265437 12.996224
##  [22] 12.723993 12.450794 12.178207 11.907451 11.639453 11.374916 11.114365
##  [29] 10.858182 10.606643 10.359936 10.118182  9.881450  9.649767  9.423130
##  [36]  9.201511  8.984864  8.773128  8.566232  8.364096  8.166637  7.973766
##  [43]  7.785391  7.601421  7.421762  7.246320  7.075003  6.907719  6.744375
##  [50]  6.584883  6.429154  6.277101  6.128639  5.983683  5.842153  5.703967
##  [57]  5.569048  5.437319  5.308704  5.183130  5.060526  4.940822  4.823948
##  [64]  4.709839  4.598428  4.489653  4.383450  4.279760  4.178522  4.079679
##  [71]  3.983174  3.888952  3.796959  3.707141  3.619449  3.533830  3.450237
##  [78]  3.368621  3.288936  3.211136  3.135176  3.061013  2.988605  2.917909
##  [85]  2.848885  2.781495  2.715698  2.651458  2.588737  2.527500  2.467712
##  [92]  2.409338  2.352345  2.296700  2.242371  2.189328  2.137539  2.086975
##  [99]  2.037607  1.989408  1.942348
## 
## 
## [[99]]
## [[99]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[99]]$y
##   [1]  0.000000  4.687909  8.179438 10.755345 12.631008 13.971636 14.903946
##   [8] 15.525118 15.909671 16.114737 16.184107 16.151337 16.042130 15.876165
##  [15] 15.668500 15.430646 15.171395 14.897454 14.613932 14.324707 14.032722
##  [22] 13.740196 13.448797 13.159770 12.874037 12.592271 12.314958 12.042437
##  [29] 11.774938 11.512607 11.255528 11.003734 10.757224 10.515968 10.279918
##  [36] 10.049008  9.823165  9.602304  9.386337  9.175174  8.968719  8.766877
##  [43]  8.569553  8.376651  8.188078  8.003738  7.823539  7.647392  7.475205
##  [50]  7.306891  7.142364  6.981539  6.824334  6.670667  6.520460  6.373634
##  [57]  6.230113  6.089824  5.952694  5.818651  5.687627  5.559552  5.434362
##  [64]  5.311990  5.192374  5.075451  4.961162  4.849446  4.740245  4.633503
##  [71]  4.529165  4.427177  4.327485  4.230038  4.134785  4.041677  3.950666
##  [78]  3.861704  3.774746  3.689746  3.606659  3.525444  3.446057  3.368458
##  [85]  3.292607  3.218463  3.145989  3.075147  3.005901  2.938213  2.872050
##  [92]  2.807377  2.744160  2.682366  2.621964  2.562923  2.505210  2.448797
##  [99]  2.393655  2.339754  2.287067
## 
## 
## [[100]]
## [[100]]$x
##   [1]  0.00  0.24  0.48  0.72  0.96  1.20  1.44  1.68  1.92  2.16  2.40  2.64
##  [13]  2.88  3.12  3.36  3.60  3.84  4.08  4.32  4.56  4.80  5.04  5.28  5.52
##  [25]  5.76  6.00  6.24  6.48  6.72  6.96  7.20  7.44  7.68  7.92  8.16  8.40
##  [37]  8.64  8.88  9.12  9.36  9.60  9.84 10.08 10.32 10.56 10.80 11.04 11.28
##  [49] 11.52 11.76 12.00 12.24 12.48 12.72 12.96 13.20 13.44 13.68 13.92 14.16
##  [61] 14.40 14.64 14.88 15.12 15.36 15.60 15.84 16.08 16.32 16.56 16.80 17.04
##  [73] 17.28 17.52 17.76 18.00 18.24 18.48 18.72 18.96 19.20 19.44 19.68 19.92
##  [85] 20.16 20.40 20.64 20.88 21.12 21.36 21.60 21.84 22.08 22.32 22.56 22.80
##  [97] 23.04 23.28 23.52 23.76 24.00
## 
## [[100]]$y
##   [1]  0.000000  4.481130  7.824977 10.293878 12.090265 13.370330 14.254570
##   [8] 14.835930 15.186090 15.360316 15.401203 15.341570 15.206688 15.016005
##  [15] 14.784475 14.523582 14.242139 13.946891 13.642994 13.334376 13.024018
##  [22] 12.714173 12.406532 12.102354 11.802566 11.507839 11.218647 10.935315
##  [29] 10.658054 10.386985 10.122164  9.863596  9.611248  9.365059  9.124947
##  [36]  8.890814  8.662552  8.440045  8.223172  8.011809  7.805830  7.605109
##  [43]  7.409521  7.218941  7.033245  6.852313  6.676025  6.504264  6.336916
##  [50]  6.173870  6.015015  5.860244  5.709454  5.562541  5.419408  5.279957
##  [57]  5.144094  5.011725  4.882763  4.757119  4.634707  4.515445  4.399252
##  [64]  4.286049  4.175759  4.068306  3.963619  3.861625  3.762256  3.665444
##  [71]  3.571123  3.479229  3.389700  3.302474  3.217494  3.134699  3.054036
##  [78]  2.975448  2.898882  2.824287  2.751611  2.680805  2.611821  2.544612
##  [85]  2.479133  2.415339  2.353186  2.292633  2.233638  2.176161  2.120163
##  [92]  2.065605  2.012452  1.960667  1.910214  1.861059  1.813170  1.766512
##  [99]  1.721056  1.676768  1.633621

Example 2: multi-factor experiments to build (hierarchical) logistic regression models for pharmaceutical salt formation

Four controllable variables:

• rate of agitation during mixing (\(x_1\))
• volume of composition (\(x_2\))
• temperature (\(x_3\))
• evaporation rate (\(x_4\))

For the \(j\)th observation in the \(i\)th group \((i=1,\ldots,g;\, j=1,\ldots,n_g)\): \[ y_{ij} \sim \mbox{Bernoulli}\left(\rho(\boldsymbol{x}_{ij})\right) \] with \[ \log\left(\frac{\rho(\boldsymbol{x}_{ij})}{1-\rho(\boldsymbol{x}_{ij})}\right) = \left(\beta_0 + \omega_{i0}\right) + \sum_{r=1}^k\left(\beta_r + \omega_{ir}\right)x_{ijr}\,, \] where \(x_{ijr}\) is the value taken by the \(r\)th variable.

• \(\boldsymbol{\beta}= (\beta_0,\beta_1,\ldots,\beta_{q-1})^\mathrm{T}\) are unknown parameters of interest
• \(\boldsymbol{\omega}_i = (\omega_{i0}, \omega_{i1}, \ldots, \omega_{iq-1})^\mathrm{T}\) are group specific parameters for the \(i\)th group

Prior distributions (for later use):

• \(\beta_0 \sim U(-3,3)\), \(\beta_1 \sim U(4, 10)\), \(\beta_2 \sim U(5, 11)\), \(\beta_3 \sim U(-6, 0)\), \(\beta_4 \sim U(-2.5, 3.5)\)

1. standard logistic regression - \(\omega_{ir} = 0\)
2. hierarchical logistic regression - \(\omega_{ir} \sim U(-s_r, +s_r)\). with \(s_{r}>0\) following a triangular distribution

Many Frequentist criteria for finding optimal designs for both linear and nonlinear models optimise a function of the information matrix; see Atkinson, Donev, and Tobias (2007), ch.10

• we have already seen \(D\)-optimality

Let \(\xi = (\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)^\mathrm{T}\) denote a design, and set \(M(\xi;\,\boldsymbol{\theta}) = M(\boldsymbol{\theta})\) to explicitly acknowledge the dependence of the information matrix on the design

• \(D\)-optimality: maximise \(\phi_D(\xi) = \mbox{det}\, M(\xi;\,\boldsymbol{\theta})\)
• \(A\)-optimality: minimise \(\phi_A(\xi) = \mbox{trace}\, M(\xi;\,\boldsymbol{\theta})^{-1}\)
• \(G\)-optimality: minimise \(\phi_G(\xi) = \max_\boldsymbol{x}\mbox{Var}(\hat{y}(\boldsymbol{x}))\)
  • where \(\hat{y}(x)\) is the predicted response at \(\boldsymbol{x}\) and the (asymptotic) prediction variance is a function of \(M(\xi;\,\boldsymbol{\theta})\)
• \(V\)- (or \(I\)-) optimality - minimise \(\phi_V(\xi) = \int_\mathcal{X} \mbox{Var}\left(\hat{y}(\boldsymbol{x})\right)\,\mathrm{d}\boldsymbol{x}\)

For most nonlinear models, \(M(\xi;\,\boldsymbol{\theta})\) will be a function of the unknown parameters \(\boldsymbol{\theta}\) (unlike for the linear model, where \(M(\xi;\,\boldsymbol{\beta}) = X^\mathrm{T}X\))

This leads to a “chicken and egg” situation

• if you can tell me the values of the unknown parameters, I can give you an optimal design
• but if you knew the value of \(\boldsymbol{\theta}\), you probably wouldn’t need to perform the experiment!

For some models/experiments, the quality of a design may change a lot with the value of \(\boldsymbol{\theta}\)

rho <- function(x, beta0 = 0, beta1 = 1) {
  eta <- beta0 + beta1 * x
  1 / (1 + exp(-eta))
}
par(mar = c(8, 4, 1, 2) + 0.1)
curve(rho, from = -5, to = 5, ylab = expression(rho), xlab = expression(italic(x)), cex.lab = 1.5, 
      cex.axis = 1.5, ylim = c(0, 1), lwd = 2)

For simple logistic regression, the information matrix has the form \[ M(\xi;\,\boldsymbol{\beta}) = X^\mathrm{T}W X\,, \] with \(X\) the \(n\times 2\) model matrix and \(W = \mbox{diag}\left\{\rho(x_i)[1-\rho(x_i)]\right\}\)

For example with \(n=2\), \(\xi = (-1, 1)\), \(\beta_0=0\) and \(\beta_1 = 1\) \[ M(\xi;\,\boldsymbol{\beta}) = \left( \begin{array}{cc} 1 & 1 \\ -1 & 1 \end{array} \right) \left( \begin{array}{cc} 0.2 & 0 \\ 0 & 0.2 \end{array} \right) \left( \begin{array}{cc} 1 & -1 \\ 1 & 1 \end{array} \right) \]

Minfo <- function(xi, beta0 = 0, beta1 = 1) {
  X <- cbind(c(1, 1), xi)
  v <- function(x) rho(x, beta0, beta1) * (1 - rho(x, beta0, beta1))
  W <- diag(c(v(xi[1]), v(xi[2])))
  t(X) %*% W %*% X
}
Dcrit <- function(xi, beta0 = 0, beta1 = 1) {
  d <- det(Minfo(xi, beta0, beta1))
  ifelse(is.nan(d), -Inf, d)
}

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 1\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1))
xi.opt1 <- dopt$par
xi.opt1

## [1] -1.543421  1.543530

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 2\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = 2)
xi.opt2 <- dopt$par
xi.opt2

## [1] -0.7717705  0.7717418

Locally \(D\)-optimal designs

\(\beta_0 = 0, \beta_1 = 0.5\)

dopt <- optim(par = c(-1, 1), Dcrit, control = list(fnscale = -1), beta1 = .5)
xi.opt3 <- dopt$par
xi.opt3

## [1] -3.086913  3.086981

Getting \(\beta_1\) wrong: design for \(\beta_1 = .5\) when actually \(\beta_1 = 2\)

\(D\)-efficiency

(Dcrit(xi.opt3, beta1 = 2) / Dcrit(xi.opt2, beta1 = 2)) ^ (1 / 2)

## [1] 0.05720905

Use of the “wrong” design can lead to uninformative experiments (with “small” information matrices)

For the logistic regression example, the drop in efficiency is closely related to the phenomenon of separation (see Firth 1993)

Motivates the need for designs which are robust to the values of the model parameters

• maximin designs (focus on worst case performance)
• Bayesian designs

Decision-theoretic design starts with a utility function \(u(\xi,\boldsymbol{y},\boldsymbol{\theta})\) that defines the usefulness of a design for a particular purpose, given data \(\boldsymbol{y}\) and parameters \(\boldsymbol{\theta}\)

Common choices of utility function include

• negative squared error loss \[u(\xi, \boldsymbol{y}, \boldsymbol{\theta}) = -\left[\boldsymbol{\theta}- E(\boldsymbol{\theta}\,|\,\boldsymbol{y})\right]^2\]
  • negative squared difference between \(\boldsymbol{\theta}\) and the posterior mean
• surprisal or self information \[ \begin{split} u(\xi, \boldsymbol{y}, \boldsymbol{\theta}) & = \log \pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi) - \log \pi(\boldsymbol{\theta}) \\ & = \log \pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi) - \log \pi(\boldsymbol{y}\,|\,\xi) \end{split} \]
  • difference between log posterior and log prior densities, or between the log-likelihood and the log-evidence

A priori (before the experiment), we do not know \(\boldsymbol{y}\) or \(\boldsymbol{\theta}\) (we will never know \(\boldsymbol{\theta}\))

So, we take the expectation of the utility function with respect to the joint distribution of \(\boldsymbol{y},\boldsymbol{\theta}\)

\[ \begin{split} U(\xi) & = E_{\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi}\left[u(\xi,\boldsymbol{y},\boldsymbol{\theta})\right]\\ & = \int u(\xi,\boldsymbol{y},\boldsymbol{\theta})\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int u(\xi, \boldsymbol{y}, \boldsymbol{\theta})\pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)\pi(\boldsymbol{y}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int u(\xi, \boldsymbol{y}, \boldsymbol{\theta})\pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi)\pi(\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y} \end{split} \] The equivalence of the third and fourth equations follows from Bayes theorem

• the third equation more clearly shows the dependence on the posterior distribution
• the fourth equation is often more useful for calculations and computation

See Chaloner and Verdinelli (1995)

Surprisal \[ \begin{split} U(\xi) & = \int \log \frac{\pi(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)}{\pi(\boldsymbol{\theta})}\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = \int \log \frac{\pi(\boldsymbol{y}\,|\,\boldsymbol{\theta},\xi)}{\pi(\boldsymbol{y}\,|\,\xi)}\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y} \end{split} \] - the expected Shannon information gain (SIG) or expected Kullback-Liebler divergence between prior and posterior densities

Negative squared error loss \[ \begin{split} U(\xi) & = - \int \left[\boldsymbol{\theta}- E(\boldsymbol{\theta}\,|\,\boldsymbol{y})\right]^2\pi(\boldsymbol{y},\boldsymbol{\theta}\,|\,\xi)\,\mathrm{d}\boldsymbol{\theta}\,\mathrm{d}\boldsymbol{y}\\ & = - \int \mbox{tr}\left\{\mbox{Var}(\boldsymbol{\theta}\,|\,\boldsymbol{y},\xi)\pi(\boldsymbol{y}\,|\,\xi)\right\}\,\mathrm{d}\boldsymbol{y} \end{split} \] - the expected negative squared error loss (NSEL)

In general, Bayesian design is easy in principle but hard in practice

1. For most nonlinear models, the expected utility will be intractable and involves high-dimensional integrals with respect to \(\boldsymbol{y}\) - often, obtaining the utility function itself requires the solution of intractable integrals (cf both ESIG and NSEL) - numerical or analytical approximation is required (eg Ryan et al. 2016)

2. A high-dimensional optimisation problem results for multi-factor experiments with many design points

For large \(n\), the inverse information matrix \(M(\xi;\,\boldsymbol{\theta})\) is an asymptotic approximation to the posterior variance-covariance matrix

Using this approximation, we can define Bayesian analogues of classical optimality criteria

\(D\)-optimality: maximise \[ U_D(\xi) = \int \log\mbox{det} M(\xi;\,\boldsymbol{\theta})\pi(\boldsymbol{\theta})\,\mathrm{d}\boldsymbol{\theta} \]

• approximation to ESIG

\(A\)-optimality: maximise \[ U_A(\xi) = - \int \mbox{tr} M^{-1}(\xi;\,\boldsymbol{\theta})\pi(\boldsymbol{\theta})\,\mathrm{d}\boldsymbol{\theta} \]

• approximation to NSEL

These integrals, with respect to \(\boldsymbol{\theta}\), are lower dimensional and more amenable to deterministic (quadrature) approximation, eg Gotwalt, Jones, and Steinberg (2009)

The acebayes package provides functions for constructing approximations to expected utilities

• default is to use quadrature to approximate the Bayesian \(D\)-optimality objective function

library(acebayes)
prior <- list(support = matrix(c(0, 0, .5, 2), nrow = 2))
logreg.util <- utilityglm(formula = ~ x, family = binomial, prior = prior)$utility
BDcrit <- function(xi) logreg.util(data.frame(x = xi))
bdopt <- optim(par = c(-1, 1), BDcrit, control = list(fnscale = -1))
bdopt$par

## [1] -1.202960  1.203132

As an alternative to analytical approximations, Monte Carlo approximation to the expected utility is simple to implement and intuitively appealing

\[ \tilde{U}(\xi) = \frac{1}{B}\sum_{i=1}^B\tilde{u}(\xi, \boldsymbol{y}_i, \boldsymbol{\theta}_i) \] where

• \(\left\{\boldsymbol{\theta}_h, \boldsymbol{y}_h\right\}_{h=1}^B\) is a random sample from \(\pi(\boldsymbol{\theta},\boldsymbol{y}\,|\,\xi)\)
• \(\tilde{u}(\xi,\boldsymbol{y},\boldsymbol{\theta})\) is, where necessary, an approximation to the utility function (often, nested Monte Carlo is required)

How to construct the approximation \(\tilde{u}(\xi,\boldsymbol{y},\boldsymbol{\theta})\) is an active area of research, eg Overstall, McGree, and Drovandi (2018), Beck et al. (2018)

Find an optimal design using Monte Carlo:

priorMC <- function(B) cbind(rep(0, B), runif(n = B, min = .5, max = 2))
logreg.utilSIG <- utilityglm(formula = ~ x, family = binomial, prior = priorMC, criterion = "SIG")$utility
BDcritSIG <- function(xi, B = 1000) mean(logreg.utilSIG(data.frame(x = xi), B))
bdoptSIG <- optim(par = c(-1, 1), BDcritSIG, control = list(fnscale = -1))
bdoptSIG$par

## [1] -1.215172  1.197912

bdopt$par

## [1] -1.202960  1.203132

Larger Monte Carlo sample sizes will produce results more similar to the design found using quadrature (in this example)

In general, direct optimisation of the Monte Carlo approximation requires large \(B\) to generate suitable smooth objective function and/or expensive stochastic algorithms (eg genetic algorithms)

Hamada et al. (2001)

Alternatively, the optimisation can be embedded within a simulation scheme and samples generated from the joint artificial distribution of \(\xi,\boldsymbol{y},\boldsymbol{\theta}\)

• take \(\xi^*\), the optimal design, to be the posterior mode of the marginal distribution
• most effective for small experiments (both numbers of variables and runs)

Müller (1999), Müller, Sansó, and De Iorio (2004)

Instead of directly minimising a Monte Carlo approximation to the expected utility, find designs via curve fitting (Müller and Parmigiani 1996)

1. Evaluate the Monte Carlo approximation \(\tilde{U}(\xi)\) for a small number of designs, \(\xi_1,\ldots,\xi_Q\)
2. Smooth the “data” \(\left\{\xi_i, \tilde{U}(\xi_i)\right\}\), i.e. fit a statistical model, to obtain a surrogate \(\hat{U}(\xi)\)
3. Find \(\xi\) that maximises \(\hat{U}(\xi)\)

Return to Example 1, compartmental model

• find a design with \(n=2\) runs, with fixed \(x_1 = 5\)
• use Monte Carlo approximation to SIG for 10 values of \(x_2\)

library(DiceKriging)
library(DiceDesign)
n <- 10; x1<- -0.583; x2 <- 2 * maximinSA_LHS(lhsDesign(n, 1)$design)$design- 1
u <- NULL; for(i in 1:n) u[i] <- mean(utilcomp15sig(c(x1, x2[i]), B = 1000))
par(mar = c(4, 4, 2, 2) + 0.1)
plot(12 * (x2 + 1), u, xlab = expression(x[2]), ylab = "Approx. expected SIG", xlim = c(0, 24), 
     ylim = c(0, 2), pch = 16, cex = 1.5); abline(v = 12 * (x1 + 1), lwd = 2)
usmooth <- km(design = 12 * (x2 + 1), response = u, nugget = 1e-3, control = list(trace = F))
xgrid <- matrix(seq(0, 24, l = 1000), ncol = 1); pred <- predict(usmooth, xgrid, type = "SK")$mean
lines(seq(0, 24, l = 1000), pred, col = "blue", lwd = 2); abline(v = xgrid[which.max(pred), ], lty = 2)

Coordinate exchange, a version of cyclic ascent, is a popular algorithm for finding optimal designs (Meyer and Nachtsheim 1995)

• optimisation of \(\xi = (\boldsymbol{x}_1,\ldots,\boldsymbol{x}_n)\) proceeds coordinate-wise, i.e. just one of the \(x_{ij}\) is varied at a time

Approximate coordinate exchange (ACE) combines coordinate exchange with smoothing to find high-dimensional designs under computationally expensive approximate expected utilities

• a nonparametric regression model (a Gaussian process) is used to smooth the Monte Carlo approximations of \(U(\xi)\) as a function of one coordinate
  • reduces the computational burden
  • facilitates optimisation of a noisy function

Overstall and Woods (2017)

Return to the multifactor logistic regression (crystallography) example

## set up prior
priorMFL <- function(B) {
  b0 <- runif(B, -3, 3)
  b1 <- runif(B, 4, 10)
  b2 <- runif(B, 5, 11)
  b3 <- runif(B, -6, 0)
  b4 <- runif(B, -2.5, 2.5)
  cbind(b0, b1, b2, b3, b4)
}
## define the utility function
MFL.utilSIG <- utilityglm(formula = ~ x1 + x2 + x3 + x4, family = binomial, prior = priorMFL, 
                          criterion = "SIG")$utility
## starting design with n=18 runs, on [-1, 1]
d <- 2 * randomLHS(18, 4) - 1
colnames(d) <- paste0("x", 1:4)
## approximate expected utility for starting design
mean(MFL.utilSIG(d, 1000))

## [1] 1.309741

## not run - quite computationally expensive
MLF.ace <- ace(utility = MFL.utilSIG, start.d = d, progress = T)

For this logistic regression example, acebayes has some designs precomputed

pairs(optdeslrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])), cex = 2)

Hierachical logistic regression with \(g=3\) groups (blocks, eg wellplates)

pairs(optdeshlrsig(18), pch = 16, 
      labels=c(expression(x[1]), expression(x[2]), expression(x[3]), expression(x[4])),
      col = c("black", "red", "blue")[rep(1:3, rep(6, 3))], cex = 2)

Many physical and social processes can be approximated by computer codes which encapsulate mathematical models

• eg partial differential equations solved using finite element methods
• eg reaction kinetics modelling in computational biology, in-silico chemistry

- computer code: numerical implementation of the mathematical model

Key feature: the model does not have a closed-form; it can only be evaluated numerically, and this is typically (relatively) expensive

We will focus on deterministic computer models

Assumption: \(g(\boldsymbol{x})\) can only be evaluated numerically; i.e. \(g(\boldsymbol{x})\) can be computed for a given \(\boldsymbol{x}\) but the general form is unknown

How do we learn about the function \(g(\boldsymbol{x})\)?

In an analogy to a physical system, we experiment on \(g(\boldsymbol{x})\), i.e.

• choose a design \(\xi = (\boldsymbol{x}_1,\ldots, \boldsymbol{x}_n)\)
• evaluate \(g(\boldsymbol{x}_i)\) (run the computer code)

Use the “data” \(\left\{\boldsymbol{x}_i, g(\boldsymbol{x}_i)\right\}\) to build statistical models linking \(\boldsymbol{x}\) and \(g(\boldsymbol{x})\)

• called emulators; typically use a Gaussian process

See Santner, Williams, and Notz (2003)

Climate modelling involves the solution of many intractable equations, leading to mathematical models evaluated via computationally expensive computer codes

• lots of applications of computer experiments

We will illustrate methods on a very simple example: a time-stepping advective/diffusive surface layer meridional EBM (energy balance model)

• 2D earth with no land
• each surface object has a percentage of ice cover
• different albedo (fraction of solar energy reflected) for ice vs non-ice surfaces
• ocean circulation is explicitly modelling (cf Atlantic gulf stream)
• two variables: \(x_1\) - solar constant; \(x_2\) - non-ice albedo
• output is mean temperature

See https://wiki.aston.ac.uk/foswiki/bin/view/MUCM/SurfebmModel

## design and data are in 'ebm'
library(akima)
fld <- interp(x = ebm$x1, y = ebm$x2, z = ebm$y)
filled.contour(x = fld$x, y = fld$y, z = fld$z, asp = 1)

As we will see later, emulators are usually constructed using nonparametric statistical models

This choice leads naturally to using space-filling designs

• such designs do not rely on the functional form of the relationship between the code inputs and the response
• good coverage is important for prediction (we will predict “better” near points we have already run the computer model)

Common designs are chosen to optimise some space-filling metric, or formed from (stratified) random sampling

Space-filling designs do not have replication, so ideal for deterministic computer models

Many designs proposed for computer experiments are related to ideas underpinning quadrature, and the approximation of an expectation.

Let \(\bar{g} = \frac{1}{n}\sum_{i=1}^n g(\boldsymbol{x}_i)\), the sample mean of \(g(\cdot)\) for \(\xi\). Then

\[ |E_\boldsymbol{x}[g(\boldsymbol{x})] - \bar{g}| \le \mbox{constant}\times D(\xi) \] where \(D(\xi)\) is the star discrepancy of the design

• \(D(\xi)\) is a measure of the uniformity of the design points

This relationship leads to the criterion of design selection via minimising discrepancy

• \(D(\xi)\) is difficult to compute for moderate to high numbers of dimensions
• therefore, it is more common to minimise the related centred \(L_2\)-discrepancy

Fang, Li, and Sudjianto (2006), Ch.3

Two sensible criteria for the selection of a space-filling design are

• make sure no two points in the design are too close together
• make sure no point in the design region is too far from a design point

(Johnson, Moore, and Ylvisaker 1990)

The Euclidean distance between points \(\boldsymbol{x}\) and \(\boldsymbol{x}^\prime\) is given by

\[ \delta(\boldsymbol{x}, \boldsymbol{x}^\prime) = \sqrt{\sum_{i=1}^k \left(x_{j} - x^\prime_j\right)^2} \]

Using Euclidean distance, we can define

• maximin (Mm) criterion: maximise

\[ \min_{\boldsymbol{x}_i, \boldsymbol{x}_j\in\xi}\delta(\boldsymbol{x}_i, \boldsymbol{x}_j) \]

• minimax (mM) criterion: minimise

\[ \max_{\boldsymbol{x}}\delta(\boldsymbol{x}, \xi) \] where the distance between a point \(\boldsymbol{x}\) and a design \(\xi\) is defined as

\[ \delta(\boldsymbol{x}, \xi) = \min_{\boldsymbol{x}_j\in\xi}\delta(\boldsymbol{x}, \boldsymbol{x}_i) \]

Roughly speaking, an Mm design spreads out the design points, and an mM design covers the design region

Intuitively, covering the design region seems more desirable (eg for prediction), but optimising the mM objective function is computationally challenging. Hence, Mm designs are more commonly used

For high-dimensional problems, space-filling is difficult

• many points are required to adequate space-fill a high-dimensional space (curse of dimensionality)

Latin hypercube designs (LHDs) are randomly chosen sets of points with the restriction of uniform one-dimensional projections (McKay, Beckman, and Conover 1979)

• each variable has no overlapping points, and good coverage (compare with a factorial design, which has hidden replication)
• can be easily constructed using permutations of integers

An LHD only guarantees space-filling properties in each one-dimensional projection, not overall. So we normally combine the Latin hypercube principle with a space-filling criteria, eg to find a Mm LHD

LH <- function(n = 3, d = 2) {
    D <- NULL
    for(i in 1:d) D <- cbind(D, sample(1:n, n))
    D 
}
set.seed(4)
par(mar=c(5,6,2,4)+0.1, pty = "s")
plot((LH() -.5)/ 3, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2)
abline(v = c(0, 1/3, 2/3, 1), lty = 2)
abline(h = c(0, 1/3, 2/3, 1), lty = 2)

The DiceDesign package has functions to generate various LHDs

library(DiceDesign)
lhs.d <- lhsDesign(9, 2)
plot(lhs.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "random", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

discrep.d <- discrepSA_LHS(lhs.d$design, criterion = "C2")
plot(discrep.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "discrepancy", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

maximin.d <- maximinSA_LHS(discrep.d$design)
plot(maximin.d$design, xlim = c(0, 1), ylim = c(0, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, cex.lab = 2, cex.axis = 2, cex = 2, 
     main = "maximin", cex.main = 2)
abline(v = seq(0, 9) / 9, lty = 2)
abline(h = seq(0, 9) / 9, lty = 2)

The design for the EBM example is a Mm LHD

par(mar=c(5,6,2,4)+0.1, pty = "s")
plot(ebm[, 2:3], xlim = c(-1, 1), ylim = c(-1, 1), pty = "s", xlab = expression(x[1]), 
     ylab = expression(x[2]), pch = 16, asp = 1)
abline(v = 2 * seq(0:20) / 20 - 1, lty = 2)
abline(h = 2 * seq(0:20) / 20 - 1, lty = 2)

The most common statistical model used to emulate computer models is the Gaussian process (GP)

• flexible, nonparametric regression model (few assumptions made about \(g(\boldsymbol{x})\))
• naturally allows for uncertainty quantification (eg prediction intervals)
• interpolates observed responses

An intuitive way to think about a GP is as a prior for the unknown function \(g(\boldsymbol{x})\) within a Bayesian framework

We say that

\[ g(\boldsymbol{x})\sim \text{GP}\left(\boldsymbol{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}, \sigma^2\kappa(\boldsymbol{x},\boldsymbol{x}^\prime;\,\boldsymbol{\theta})\right)\,, \] where \(\boldsymbol{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}\) is the mean, \(\kappa(\boldsymbol{x},\boldsymbol{x}^\prime;\,\boldsymbol{\phi})\) is the correlation function, \(\boldsymbol{\theta}\) is the vector of correlation parameters and \(\sigma^2\) is the constant variance, if:

• any vector \(\boldsymbol{g}= \left(g(\boldsymbol{x}_1), \dots , g(\boldsymbol{x}_n)\right)^{\mathrm{T}}\) satisfies \[\boldsymbol{g}\sim N\left(F\boldsymbol{\beta}, \sigma^2 K(\boldsymbol{\theta})\right)\,,\] with \(F\) a model matrix and \(K\) the \(m\times m\) covariance matrix defined by \(K(\boldsymbol{\theta})_{ij} = \kappa(\boldsymbol{x}_i,\boldsymbol{x}_j;\boldsymbol{\theta})\).

See Rasmussen and Williams (2006)

Typically, very simple mean functions are chosen for the GP, eg

• constant: \(\boldsymbol{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}= \beta_0\) (sometimes called ordinary kriging)
• linear: \(\boldsymbol{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}= \beta_0 + \sum_{j=1}^k\beta_jx_j\) (universal kriging)

The most commonly used correlation functions are separable and stationary

• squared exponential:

\[ \kappa(\boldsymbol{x}, \boldsymbol{x}^\prime;\,\boldsymbol{\theta})=\exp\left[-\sum_j\left(\frac{|x_{j} - x^\prime_{j} |}{\theta_j }\right)^2\right] \]

• Matérn \(\nu = 5/2\)

\[ \kappa(\boldsymbol{x}, \boldsymbol{x}^\prime; \,\boldsymbol{\theta}) = \prod_{j}\left(1 + \sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j} + \frac{5}{3}\left(\frac{|x_j - x_j^\prime|}{\theta_j}\right)^2\right)\exp\left(-\sqrt{5}\frac{|x_j - x_j^\prime|}{\theta_j}\right) \] The Matérn function can be defined for other values of \(\nu\); for \(\nu\rightarrow\infty\), the squared exponential function is obtained

Given model evaluations \(\boldsymbol{g}= \left[g(\boldsymbol{x}_1), \ldots, g(\boldsymbol{x}_n)\right]\), a posterior GP can be obtained:

\[ g(\boldsymbol{x})\,|\, \boldsymbol{g},\boldsymbol{\beta},\boldsymbol{\theta},\sigma^2 \sim N\left(m(\boldsymbol{x}), s^2(\boldsymbol{x})\right) \]

• \(m(\boldsymbol{x}) = \boldsymbol{f}(\boldsymbol{x})^\mathrm{T}\boldsymbol{\beta}+ \boldsymbol{\kappa}_n^\mathrm{T}K^{-1}(\boldsymbol{g}- F\boldsymbol{\beta})\)
• \(s^2(\boldsymbol{x}) = \sigma^2\left(1 - \boldsymbol{\kappa}_n^\mathrm{T}K^{-1}\boldsymbol{\kappa}_n\right)\)

where \(\boldsymbol{\kappa}_n = [\kappa(\boldsymbol{x},\boldsymbol{x}_i\,;\,\boldsymbol{\theta})]_{i=1}^n\) is a vector of correlations between \(g(\boldsymbol{x})\) and \(g(\boldsymbol{x}_1),\ldots,g(\boldsymbol{x}_n)\)

The updating of the prior mean and variance depends on the “distance” between \(\boldsymbol{x}\) and the points in \(\xi\)

• the posterior mean will be adjusted more for points closer to the design
• predictions at these points will have smaller posterior variance

If \(\boldsymbol{x}= \boldsymbol{x}_i\) (so we are predicting at a design point), \(K^{-1}\boldsymbol{\kappa}_n = \boldsymbol{e}_i\), the \(i\)th unit vector

• \(m(\boldsymbol{x}_i) = \boldsymbol{f}(\boldsymbol{x}_i)^\mathrm{T}\boldsymbol{\beta}+ \boldsymbol{e}_i^\mathrm{T}(\boldsymbol{g}- F\boldsymbol{\beta}) = g(\boldsymbol{x}_i)\)
• \(s^2(\boldsymbol{x}_i) = \sigma^2\left(1 - \boldsymbol{\kappa}_n^\mathrm{T}\boldsymbol{e}_i\right) = \sigma^2\left(1 - \kappa(\boldsymbol{x}_i,\boldsymbol{x}_i\,;\,\boldsymbol{\theta})\right) = 0\)

The posterior GP interpolates - exactly what you want for a deterministic computer code

Inference unconditional on all the hyperparameters requires numerical approximation (eg Markov chain Monte Carlo)

• it is common to estimate the parameters, eg using maximum likelihood, to “plug-in” to the posterior predictive distribution

A simple example: \(g(x) = \sin(2\pi x)\) using the DiceKriging package

library(DiceDesign)
library(DiceKriging)
xi <- lhsDesign(6, 1)$design
y <- sin(2 * pi * xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
legend(x = "topright", legend = c("posterior mean of g", "posterior quantiles for g", 
                                  expression(paste("observed data ", g(x[i])))), lty = c(1, 2, NA), 
       pch = c(NA, NA, 4), lwd = c(4, 4, 4), col = c("red", "black", "blue"))

Return to the EBM example

gpebm <- km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
gpebm

## 
## Call:
## km(formula = ~., design = ebm[, 2:3], response = ebm[, 1], control = list(trace = F))
## 
## Trend  coeff.:
##                Estimate
##  (Intercept)    16.3262
##           x1     2.4078
##           x2   -28.9973
## 
## Covar. type  : matern5_2 
## Covar. coeff.:
##                Estimate
##    theta(x1)     2.8829
##    theta(x2)     0.2722
## 
## Variance estimate: 2.215036

xs1 <- sort(c(seq(-1, 1, length = 10), ebm[, 2]))
xs2 <- sort(c(seq(-1, 1, length = 10), ebm[, 3]))
xs <- expand.grid(x1 = xs1, x2 = xs2)
gppebm <- predict(gpebm, newdata = xs, type = "UK")
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$mean, nrow = length(xs1)))
filled.contour(x = xs1, y = xs2, z = matrix(gppebm$sd, nrow = length(xs1)))

A common task is optimisation of \(g(\boldsymbol{x})\)

When \(g(\boldsymbol{x})\) is computationally expensive to evaluate, computer experiments and emulators can be used to facilitate the optimisation.

The field of Bayesian optimisation uses sequentially collected evaluations of \(g(\boldsymbol{x})\)

• place a prior distribution (eg GP) on \(g(\boldsymbol{x})\)
• collect function evaluations at points chosen sequentially via an acquisition function
• update the prior to a posterior distribution, and infer the maximum/minimum of \(g(\boldsymbol{x})\)

Uncertainty in the posterior (i.e. for \(g(\boldsymbol{x})\) at unobserved \(\boldsymbol{x}\)) leads to exploration/exploitation trade-off

The most common acquisition function is expected improvement (EI)

See Jones, Schonlau, and Welch (1998)

For a deterministic computer model and a minimisation problem, the improvement from performing one more run is given by: \[ \max(g_\min - g(\boldsymbol{x}), 0) \] where \(g_\min\) is the minimum across the model runs performed to date

This quantity is a random variable - we are uncertain about \(g(\boldsymbol{x})\) at a point we have not observed.

EI chooses \(\boldsymbol{x}\) to maximise \[ E_g\left[\max(g_\min - g(\boldsymbol{x}), 0)\,;\, \boldsymbol{g}\right] = \left[g_\min - m(\boldsymbol{x})\right]\Phi\left(\frac{g_\min - m(\boldsymbol{x})}{s(\boldsymbol{x})}\right) + s(\boldsymbol{x})\phi\left(\frac{g_\min - m(\boldsymbol{x})}{s(\boldsymbol{x})}\right) \] where \(\phi\) and \(\Phi\) are the standard normal pdf and cdf, respectively

EI is an decreasing function of \(m(\boldsymbol{x})\) and an increasing function of \(s^2(\boldsymbol{x})\), so it leads to choosing design points that either minimise the posterior mean or the posterior variance

• experiment either where our uncertainty is high or near where we predict the minimum to be (explore or exploit)

A simple example: \(g(\boldsymbol{x}) = \sin(2\pi x)\) but with a different starting design using DiceOptim

xi <- matrix(c(0.1, 0.8, 0.9), ncol = 1)
fn <- function(x) sin(2 * pi * x)
y <- fn(xi)
gp <- km(design = xi, response = y, control = list(trace = F))
xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp, newdata = xs, type = "SK")

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x")
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)

library(DiceOptim)
xin <- max_EI(model = gp, lower = 0, upper = 1)$par

## 
## 
## Wed Sep  4 10:52:36 2019
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  8.397787e-02
##       1  1.140734e-01
##       2  1.180704e-01
##       3  1.211283e-01
##       4  1.211283e-01
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 1.211283e-01
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 6.719061e-01    G[ 1] : -3.212443e-09
## 
## Solution Found Generation 4
## Number of Generations Run 7
## 
## Wed Sep  4 10:52:36 2019
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp)

## [1] 0.1211283

plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp2 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp2, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Wed Sep  4 10:52:36 2019
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  5.143583e-02
##       1  5.149506e-02
##       3  5.280800e-02
##       4  5.280800e-02
##       5  5.280800e-02
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 5.280800e-02
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 7.500003e-01    G[ 1] : 2.844976e-10
## 
## Solution Found Generation 5
## Number of Generations Run 8
## 
## Wed Sep  4 10:52:37 2019
## Total run time : 0 hours 0 minutes and 1 seconds

EI(xin, gp2)

## [1] 0.052808

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp2, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp2), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)

xi <- rbind(xi, xin)
y <- c(y, fn(xin))
gp3 <- km(design = xi, response = y, control = list(trace = F))
xin <- max_EI(model = gp3, lower = 0, upper = 1, control = list(trace = F))$par

## 
## 
## Wed Sep  4 10:52:37 2019
## Domains:
##  0.000000e+00   <=  X1   <=    1.000000e+00 
## 
## NOTE: The total number of operators greater than population size
## NOTE: I'm increasing the population size to 10 (operators+1).
## 
## Data Type: Floating Point
## Operators (code number, name, population) 
##  (1) Cloning...........................  0
##  (2) Uniform Mutation..................  1
##  (3) Boundary Mutation.................  1
##  (4) Non-Uniform Mutation..............  1
##  (5) Polytope Crossover................  1
##  (6) Simple Crossover..................  2
##  (7) Whole Non-Uniform Mutation........  1
##  (8) Heuristic Crossover...............  2
##  (9) Local-Minimum Crossover...........  0
## 
## HARD Maximum Number of Generations: 12
## Maximum Nonchanging Generations: 2
## Population size       : 10
## Convergence Tolerance: 1.000000e-21
## 
## Using the BFGS Derivative Based Optimizer on the Best Individual Each Generation.
## Not Checking Gradients before Stopping.
## Not Using Out of Bounds Individuals and Not Allowing Trespassing.
## 
## Maximization Problem.
## 
## 
## Generation#      Solution Value
## 
##       0  3.718141e-04
##       1  5.303926e-04
##       3  5.309279e-04
##       4  5.309279e-04
##       5  7.687468e-04
##       6  7.687468e-04
## 
## 'wait.generations' limit reached.
## No significant improvement in 2 generations.
## 
## Solution Fitness Value: 7.687468e-04
## 
## Parameters at the Solution (parameter, gradient):
## 
##  X[ 1] : 7.581597e-01    G[ 1] : -4.434226e-07
## 
## Solution Found Generation 6
## Number of Generations Run 9
## 
## Wed Sep  4 10:52:37 2019
## Total run time : 0 hours 0 minutes and 0 seconds

EI(xin, gp3)

## [1] 0.0007687468

xs <- sort(c(seq(0, 1, length = 100), xi))
gpp <- predict(gp3, newdata = xs, type = "SK")
plot(xs, gpp$mean, ylim = c(-2, 2), type = "l", col = "red", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)
points(xi, y, pch = 4,lwd = 4, col = "blue")
lines(xs, gpp$upper95, lty = 2, lwd = 3)
lines(xs, gpp$lower95, lty = 2, lwd = 3)
abline(v = xin)
plot(xs, sapply(xs, EI, model = gp3), type = "l", lwd = 3, ylab = "", xlab = "x", cex.lab = 2)

To analyse the relative importance of different model inputs, functional analysis of variance (FANOVA) can be used to decompose \(g(\boldsymbol{x})\) into an additive form (see Saltelli et al. 2008)

Assume \(x_i\sim w_i(x)\), eg the input variables may be normally distributed or uniformly distributed. Then

\[ g(\boldsymbol{x}) = \mu_0 + \sum_{i=1}^k\mu_i(\boldsymbol{x}) + \mathop{\sum\sum}\limits_{i>j}\mu_{ij}(\boldsymbol{x}) + \ldots + \mu_{12\cdots k}(\boldsymbol{x}) \] where \[ \mu_0 = \int g(\boldsymbol{x})\prod_iw(x_i)\,\mathrm{d}(\boldsymbol{x})\,,\qquad \mu_i(\boldsymbol{x}) = \int g(\boldsymbol{x})\prod_{j\ne i}w_j(x_j)\,\mathrm{d}x_j - \mu_0\,, \] and \[ \mu_{ij}(\boldsymbol{x}) = \int g(\boldsymbol{x})\prod_{l \ne i,j}w_l(x_l)\,\mathrm{d}x_l - \mu_i(\boldsymbol{x}) - \mu_j(\boldsymbol{x}) + \mu_0 \] with higher-order terms defined similarly

A (normalised measure) of the impact of each variable can be obtained by assessing the variance of each \(\mu_i\) term via a Sobol’ index: \[ S_i = \frac{\mbox{Var}_{x_i}\left\{\mu_i(\boldsymbol{x})\right\}}{\mbox{Var}_\boldsymbol{x}\left\{g(\boldsymbol{x})\right\}} \in [0,1] \] The impact of higher-order terms can be assessed via the total variance index: \[ T_i = 1 - \frac{\mbox{Var}_{\boldsymbol{x}_{(i)}}\left\{\mu_{-i}(\boldsymbol{x})\right\}}{\mbox{Var}_{\boldsymbol{x}}\left\{g(\boldsymbol{x})\right\}} \in [0,1] \] where \(\boldsymbol{x}_{(i)} = (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_k)^\mathrm{T}\) and

\[ \mu_{-i}(\boldsymbol{x}) = \int g(\boldsymbol{x})w_i(x_i)\,\mathrm{d}x_i - \mu_0\,, \]

\(S_i\) measures the “main effect” of the \(i\)th variable, and \(T_i\) measures it’s total effect including through “interactions” with other variables

• big differences between \(S_i\) and \(T_i\) suggest the \(i\)th variable interacts with other variables

The sensitivity indices can be estimated using Monte Carlo methods (essentially nested MC to estimate the expectations and variances)

• for computationally expensive models, sample from a GP emulator

Various methods are implemented in the sensitivity package

EBM example:

library(sensitivity)
d <- 2; n <- 1000
X1 <- data.frame(matrix(runif(d * n), nrow = n))
X2 <- data.frame(matrix(runif(d * n), nrow = n))
colnames(X1) <- colnames(ebm)[2:3]; colnames(X2) <- colnames(ebm)[2:3]
res <- sobolGP(model = gpebm, type = "UK", MCmethod = "sobol2002", X1, X2)
res

## 
## Method: sobol2002
## 
## Model runs: 20 
## 
## Number of GP realizations: 100 
## 
## Kriging type: UK 
## 
##      estimate   std. error  min. c.i.  max. c.i.
## S1 0.02217883 0.0006346062 0.02098896 0.02339067
## S2 0.99219856 0.0004269961 0.99135448 0.99285454
## 
##       estimate   std. error   min. c.i.   max. c.i.
## T1 0.007196291 0.0005790607 0.006168938 0.008577045
## T2 0.995482832 0.0004513322 0.994654563 0.996329676

Computer experiments are an important statistical contribution to the field of uncertainty quantification (UQ)

• interdisciplinary topic on the interface of Statistics and Applied Maths
• methodologies for taking account of uncertainties when mathematical and computer models are used to describe real-world phenomena

Space-filling designs and (GP) emulators are very general, and can be applied to a variety of black box computer models

• typically require a lot less knowledge about the model than alternative methods from numerical analysis (although at some loss of efficiency)

GP emulators can be used as priors for Bayesian calibration of computer models (Kennedy and O’Hagan 2001)

• eg learning tuning parameters (cf parameter estimation, albeit with various important nuances around interpretation and physical understanding)
• data fusion: combining computer model runs and data from real experiments